#72307189

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	C₄	#21155
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#373850
prime radical	C₂²×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1797 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)(4,18,22,24)(6,21)(9,33,36,59)(10,35)(15,43,20,52)(17,40)(19,49,51,47)(23,56,34,48)(26,55)(28,37)(30,41,61,63)(31,38)(42,57)(44,62,53,50)(45,64,54,60);(1,2,5,12)(3,14)(4,19)(6,26,21,55)(7,29,25,11)(8,17)(9,34)(10,37,35,28)(13,39)(15,44,20,53)(16,38)(18,47)(22,51)(23,36)(24,49)(27,31)(30,60,61,64)(32,40)(33,56)(41,45,63,54)(42,46)(43,62,52,50)(48,59)(57,58);(1,3,7,17,5,13,25,40)(2,8,11,14,12,32,29,39)(4,20,24,52,22,15,18,43)(6,27,28,58,21,16,37,46)(9,30,33,41,36,61,59,63)(10,38,26,57,35,31,55,42)(19,50,47,53,51,62,49,44)(23,45,56,60,34,54,48,64);(1,4,21,34,7,24,37,48,5,22,6,23,25,18,28,56)(2,9,35,51,11,33,55,49,12,36,10,19,29,59,26,47)(3,15,16,45,17,43,46,60,13,20,27,54,40,52,58,64)(8,30,31,62,14,41,42,44,32,61,38,50,39,63,57,53)

Polycyclic presentation:

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

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307299, #72320114, #72320733
maximal normal subgroup of	#72307299, #72320114, #72320733

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