#72307787

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#3253
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂	C₂×#24152
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5287 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|c^2=gj,d^2=fi,e^2=ij,f^2=ij,i^2=j,b^a=bg,c^b=cg,d^a=dh,d^b=df,d^c=dgij,e^b=ei,e^c=ej,f^b=fi,g^d=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307810, #72308106
maximal normal subgroup of	#72307810, #72308106

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