#72308268

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#21401
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#363849
prime radical	#10575347	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4323 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,12)(3,15,18,9)(4,22,28,41)(5,29,31,30)(7,14,8,34)(10,38,43,25)(11,32,13,33)(16,46,24,56)(17,52,19,53)(20,45,21,37)(23,26,58,27)(35,48,40,49)(36,55,44,54)(39,57,60,42)(47,50,59,51)(61,64,63,62);(1,2)(3,15)(4,23)(5,30)(6,12)(7,14)(8,34)(9,18)(10,39)(11,32)(13,33)(16,47)(17,53)(19,52)(20,45)(21,37)(22,26)(24,59)(25,42)(27,41)(28,58)(29,31)(35,61)(36,55)(38,57)(40,63)(43,60)(44,54)(46,50)(48,64)(49,62)(51,56);(1,3)(2,9)(4,24)(5,19)(6,18)(7,20)(8,21)(10,40)(11,44)(12,15)(13,36)(14,37)(16,28)(17,31)(22,56)(23,59)(25,48)(26,51)(27,50)(29,52)(30,53)(32,55)(33,54)(34,45)(35,43)(38,49)(39,63)(41,46)(42,64)(47,58)(57,62)(60,61);(1,4,5,25,7,26,32,60,6,28,31,38,8,27,33,39)(2,10,13,41,34,57,29,58,12,43,11,22,14,42,30,23)(3,16,19,49,20,50,55,63,18,24,17,48,21,51,54,61)(9,35,36,56,45,64,52,59,15,40,44,46,37,62,53,47)

Polycyclic presentation:

2^10|a^2=j,d^2=g,e^2=ij,g^2=ij,h^2=j,i^2=j,b^a=bf,c^a=cf,c^b=cf,d^a=de,d^b=dehj,d^c=dj,e^a=ei,e^b=eij,e^c=ej,f^d=fj,g^a=gij,g^b=gi,i^a=ij,i^b=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308342
maximal normal subgroup of	#72308342

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