#72307138

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	#8375
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
socle	C₂	C₂×#8375
prime radical	#10576107	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

5817 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)(3,15,19,29)(4,20)(5,24)(10,34,31,36)(13,14)(16,44,45,41)(17,42,50,49)(18,51,53,43)(23,30)(27,28)(35,59,58,54)(37,38)(46,64,61,47)(48,63,52,60)(55,62);(1,2)(3,16,19,45)(4,21)(5,25)(6,14)(7,9)(8,13)(10,18,31,53)(11,23)(12,27)(15,41,29,44)(17,47,50,64)(20,40)(22,37)(24,39)(26,38)(28,33)(30,32)(34,43,36,51)(35,60,58,63)(42,61,49,46)(48,59,52,54)(55,57)(56,62);(2,9)(10,31)(11,32)(12,33)(13,14)(16,45)(21,40)(25,39)(34,36)(35,58)(37,38)(41,44)(46,61)(47,64)(54,59)(55,62);(1,3,4,17,8,15,23,42,7,19,20,50,6,29,30,49)(2,10,11,35,14,36,21,54,9,31,32,58,13,34,40,59)(5,18,26,52,28,43,57,63,24,53,22,48,27,51,56,60)(12,16,38,46,39,44,62,64,33,45,37,61,25,41,55,47)

Polycyclic presentation:

2^10|a^2=hj,b^2=hj,d^2=e,e^2=i,f^2=ij,g^2=j,i^2=j,b^a=bj,c^b=cj,d^a=dg,d^b=df,e^a=eh,e^b=ehij,f^a=fj,f^b=fij,f^d=fh,f^e=fj,g^b=gj,g^d=ghj,g^e=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307146, #72307535
maximal normal subgroup of	#72307146, #72307535

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