#72321369

order	1024
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#13654
commutator	C₃₂	C₂³×C₄
Frattini, cosocle	C₂×C₃₂	C₂⁴
prime radical	#513315	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ #13549 ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 128 points:

(3,16)(4,9)(5,26)(7,29)(8,35)(10,39)(11,22)(12,47)(14,50)(15,46)(17,57)(18,63)(19,66)(20,68)(21,58)(23,71)(24,76)(25,33)(27,34)(28,74)(30,78)(31,86)(32,84)(36,73)(37,59)(38,60)(40,96)(41,98)(42,99)(43,95)(44,101)(45,106)(48,55)(49,104)(51,108)(52,114)(53,112)(54,70)(56,94)(61,119)(62,120)(64,122)(65,121)(67,123)(69,103)(72,92)(75,125)(77,83)(79,80)(81,90)(82,88)(85,124)(87,91)(89,118)(93,117)(97,126)(100,115)(102,116)(105,109)(107,110)(111,128)(113,127);(1,2,6,13)(3,17,21,10)(4,22,9,11)(5,12,30,51)(7,14,32,53)(8,15,33,54)(16,57,58,39)(18,61,65,40)(19,62,67,41)(20,43,38,42)(23,49,28,44)(24,69,36,45)(25,70,35,46)(26,47,78,108)(27,81,82,48)(29,50,84,112)(31,52,85,113)(34,90,88,55)(37,93,89,56)(59,117,118,94)(60,99,68,95)(63,119,121,96)(64,100,91,97)(66,120,123,98)(71,104,74,101)(72,102,83,110)(73,106,76,103)(75,111,79,105)(77,107,92,116)(80,109,125,128)(86,114,124,127)(87,126,122,115);(1,3)(2,10)(4,20)(5,27)(6,21)(7,34)(8,37)(9,38)(11,43)(12,48)(13,17)(14,55)(15,56)(16,26)(18,29)(19,35)(22,42)(23,72)(24,77)(25,67)(28,83)(30,82)(31,87)(32,88)(33,89)(36,92)(39,47)(40,50)(41,46)(44,102)(45,107)(49,110)(51,81)(52,115)(53,90)(54,93)(57,108)(58,78)(59,86)(60,74)(61,112)(62,70)(63,79)(64,73)(65,84)(66,80)(68,71)(69,116)(75,121)(76,91)(85,122)(94,114)(95,101)(96,105)(97,106)(98,109)(99,104)(100,103)(111,119)(113,126)(117,127)(118,124)(120,128)(123,125);(1,4,6,9)(2,11,13,22)(5,28,30,23)(7,24,32,36)(8,25,33,35)(10,17)(12,49,51,44)(14,45,53,69)(15,46,54,70)(16,58)(26,71,78,74)(27,82)(29,73,84,76)(31,79,85,75)(40,61)(41,62)(42,43)(47,101,108,104)(50,103,112,106)(52,111,113,105)(55,90)(56,93)(59,118)(60,68)(63,121)(66,123)(72,83)(80,86,125,124)(87,122)(97,100)(107,116)(109,127,128,114)

Polycyclic presentation:

2^10|b^2=j,d^2=f,e^2=gh,g^2=hi,h^2=ij,i^2=j,c^a=ce,c^b=cj,d^a=dj,d^b=dj,d^c=di,e^a=eg,e^c=eg,e^d=ej,f^c=fj,g^a=gh,g^c=gh,h^a=hi,h^c=hi,i^a=ij,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321422, #72322926, #72322982
maximal normal subgroup of	#72321422, #72322926, #72322982

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);