#72320439

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#12557
commutator	C₂×C₁₆	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₁₆	C₂³
socle	C₂	#126832
prime radical	C₂×#41	#102
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#131 ◁ C₄×#504 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 128 points:

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

Polycyclic presentation:

2^10|b^2=d,c^2=e,e^2=g,f^2=hi,g^2=j,h^2=ij,i^2=j,b^a=bgij,c^a=cj,c^b=cdfh,d^a=dj,e^b=ej,f^b=fh,f^c=fhj,h^b=hi,h^c=hi,i^b=ij,i^c=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320465, #72320499, #72320948
maximal normal subgroup of	#72320465, #72320499, #72320948

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.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.6*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);