#72307961

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#185645
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
prime radical	#10576944	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3671 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)(3,15,21,46)(4,22)(5,26)(6,14)(7,10)(8,13)(9,19,36,54)(11,25)(12,30)(16,44,18,51)(17,49,48,45)(20,55,32,40)(23,58)(24,39)(27,35,62,43)(28,41)(29,37)(31,42)(33,47,56,52)(34,50,64,53)(38,63)(57,60)(59,61);(1,3,6,20,7,21,8,32)(2,9,13,35,10,36,14,43)(4,18,24,48,25,16,31,17)(5,27,29,54,28,62,30,19)(11,33,39,64,22,56,42,34)(12,40,41,46,37,55,26,15)(23,50,60,52,59,53,63,47)(38,49,61,44,57,45,58,51);(2,10)(3,16,21,18)(9,33,36,56)(11,22)(12,37)(13,14)(15,44,46,51)(17,20,48,32)(19,47,54,52)(26,41)(27,53,62,50)(34,43,64,35)(38,57)(39,42)(40,49,55,45)(58,61);(3,17)(5,28)(9,34)(12,37)(15,45)(16,20)(18,32)(19,53)(21,48)(23,59)(26,41)(27,52)(29,30)(33,43)(35,56)(36,64)(38,57)(40,44)(46,49)(47,62)(50,54)(51,55)(58,61)(60,63)

Polycyclic presentation:

2^10|a^2=hj,b^2=g,c^2=hj,e^2=j,f^2=i,g^2=j,i^2=j,b^a=bf,c^a=ch,c^b=ce,d^a=dhj,d^b=dei,e^a=ej,e^b=eh,f^a=fi,f^b=fi,f^d=fj,g^a=gj,g^c=ghj,g^d=ghj,g^e=gj,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307980
maximal normal subgroup of	#72307980

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