#72287302

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#3069111
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5968 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:

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

Polycyclic presentation:

2^10|d^2=fh,f^2=j,h^2=j,b^a=bf,c^a=cg,c^b=ci,d^a=dfhj,d^b=dfh,d^c=dij,e^b=ehi,e^d=ehj,f^a=fj,f^b=fj,g^b=gj,g^d=gj,h^a=hj,h^b=hj,h^e=hj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287379
maximal normal subgroup of	#72287379

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