#72322711

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#8146
commutator	C₂²×C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×#1397	C₂³
socle	C₂	#72322700
prime radical	C₂×#41	#346
LCS	G ▷ C₂²×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#1397 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

561 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=h,c^2=f,e^2=g,f^2=h,g^2=i,i^2=j,j^2=k,b^a=bd,c^a=cdijk,c^b=ch,d^b=dk,e^c=ej,f^a=fjk,f^e=fk,g^c=gk,h^a=hk

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322743, #72323087, #72323131
maximal normal subgroup of	#72322743, #72323087, #72323131

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.9*F.10*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);