#72320755

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#535
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂	#6377674
prime radical	#6469078	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2597 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)(3,15)(4,11)(5,12)(6,13)(7,14)(8,9)(10,21)(16,43)(17,36)(18,35)(19,38)(20,37)(22,39)(23,40)(24,41)(25,30)(26,54)(27,33)(28,31)(29,32)(34,48)(42,49)(44,59)(45,58)(46,61)(47,60)(50,64)(51,57)(52,55)(53,56)(62,63);(2,9)(11,30)(12,31)(13,32)(14,33)(39,55)(40,56)(41,57);(1,3)(2,10)(4,16)(5,26)(6,18)(7,20)(8,21)(9,15)(11,34)(12,42)(13,36)(14,38)(17,29)(19,27)(22,50)(23,45)(24,47)(25,48)(28,49)(30,43)(31,54)(32,35)(33,37)(39,62)(40,59)(41,61)(44,53)(46,51)(52,63)(55,64)(56,58)(57,60);(1,4,6,23,7,24,28,52,8,25,29,53,27,51,5,22)(2,11,13,40,14,41,31,55,9,30,32,56,33,57,12,39)(3,16,18,45,20,47,49,63,21,48,17,44,19,46,26,50)(10,34,36,59,38,61,54,64,15,43,35,58,37,60,42,62);(1,5,27,29,8,28,7,6)(2,12,33,32,9,31,14,13)(3,17,20,26,21,18,19,49)(4,22,51,53,25,52,24,23)(10,35,38,42,15,36,37,54)(11,39,57,56,30,55,41,40)(16,44,47,50,48,45,46,63)(34,58,61,62,43,59,60,64)

Polycyclic presentation:

2^10|d^2=g,e^2=hi,g^2=i,i^2=j,b^a=bh,c^a=cj,c^b=cf,e^c=ehi,f^a=fj,h^c=hj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320790
maximal normal subgroup of	#72320790

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