#72307259

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	#20975
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#364394
prime radical	C₂×#24285	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3765 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:

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

Polycyclic presentation:

2^10|b^2=hj,c^2=i,d^2=e,e^2=i,g^2=i,i^2=j,b^a=bh,c^a=cf,c^b=cfij,d^a=df,d^b=dg,d^c=df,e^b=ei,f^b=fj,g^b=gi,g^c=gj,h^c=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307299, #72343936, #72343978
maximal normal subgroup of	#72307299, #72343936, #72343978

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