#72306830

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#9428
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₄×M₁₆	C₂⁴
prime radical	#60639	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1886 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)(3,13)(4,18)(5,11)(6,12)(7,10)(8,15)(9,24)(14,29)(16,41)(17,42)(19,32)(20,33)(21,34)(22,52)(23,51)(25,57)(26,39)(27,37)(28,36)(30,46)(31,47)(35,54)(38,56)(40,45)(43,62)(44,60)(48,61)(49,58)(50,53)(55,63)(59,64);(1,3)(2,8)(4,19,23,48,24,53,21,55)(5,16)(6,17)(7,15)(9,32,34,61,18,50,51,63)(10,13)(11,30)(12,31)(14,22,44,25,45,54,64,56)(20,43)(26,49)(27,47)(28,46)(29,35,60,38,40,52,59,57)(33,58)(36,41)(37,42)(39,62);(1,4,20,54,27,21,5,25,7,24,26,22,6,23,28,56)(2,9,33,52,37,51,11,38,10,18,39,35,12,34,36,57)(3,14,43,53,47,64,16,48,15,45,49,19,17,44,46,55)(8,29,58,50,42,59,30,61,13,40,62,32,31,60,41,63);(2,10)(3,15)(4,21)(5,26)(6,27)(9,34)(11,33)(14,44)(16,43)(18,51)(19,53)(20,28)(23,24)(25,56)(29,59)(30,62)(31,42)(35,52)(36,39)(40,60)(41,58)(45,64)(46,49)(61,63)

Polycyclic presentation:

2^10|b^2=g,c^2=efi,e^2=hi,f^2=i,g^2=ij,h^2=j,b^a=bj,c^a=cj,c^b=ce,d^a=dj,d^b=dgi,d^c=dgi,e^c=eh,e^d=ehi,f^b=fij,f^c=fh,f^d=fi,f^e=fj,g^c=ghij,g^d=gij,h^d=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306832, #72306844, #72306974
maximal normal subgroup of	#72306832, #72306844, #72306974

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