#72307193

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	#21529
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#368381
prime radical	C₂×#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1907 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=hj,c^2=ghi,d^2=ei,f^2=j,g^2=i,i^2=j,b^a=bh,c^a=cf,c^b=cg,d^a=dfij,d^b=dfj,d^c=df,e^c=ej,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, #72320238, #72320733
maximal normal subgroup of	#72307299, #72320238, #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.7*F.8*F.9);;
gap> SetPower(coll, 4, F.5*F.9);;
gap> SetPower(coll, 6, 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.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*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);