#72307177

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#25903
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#403320
prime radical	C₂×#3067	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4073 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,c^2=j,d^2=efgij,e^2=j,g^2=i,h^2=j,i^2=j,b^a=bfj,d^a=dh,d^b=dg,d^c=de,e^b=ej,e^c=ej,e^d=ej,f^d=fj,g^a=gj,g^b=gi,g^c=gj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307305
maximal normal subgroup of	#72307305

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