#72250066

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, socle	C₂	#6453969
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #60208 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4764 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=h,c^2=h,d^2=g,f^2=j,g^2=i,i^2=j,d^a=df,d^b=dg,d^c=di,e^a=ehj,e^d=ej,f^a=fj,f^d=fj,f^e=fj,g^b=gi,g^c=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250237, #72281944
maximal normal subgroup of	#72250237, #72281944
maximal quotient of	#72250682, #72282392

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