#72307204

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#25811
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#403374
prime radical	C₂²×#41	C₂×D₄
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3872 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,c^2=j,d^2=g,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,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.7);;
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, 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);