#72250797

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	#6346637
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁵
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4314 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)(3,13,17,38)(4,10)(5,21)(6,25)(7,9)(8,16,29,44)(11,24)(12,27)(14,37,26,35)(15,39,41,60)(18,31)(19,47)(20,50)(22,28,43,36)(23,52)(30,42,55,61)(32,53)(33,49)(34,46)(40,59,51,57)(45,63)(48,54,62,58)(56,64);(1,3,6,26,7,17,27,14)(2,8,12,36,9,29,25,28)(4,15,20,51,18,41,46,40)(5,22,24,16,23,43,53,44)(10,30,34,58,31,55,50,54)(11,35,21,13,32,37,52,38)(19,48,49,42,45,62,64,61)(33,57,47,39,56,59,63,60);(3,14,17,26)(5,23)(8,28,29,36)(11,32)(13,37,38,35)(15,40,41,51)(16,43,44,22)(19,45)(21,52)(24,53)(30,54,55,58)(33,56)(39,59,60,57)(42,62,61,48)(47,63)(49,64);(2,9)(5,23)(8,29)(10,31)(12,25)(13,38)(15,41)(19,45)(24,53)(28,36)(34,50)(35,37)(40,51)(42,61)(48,62)(49,64);(1,4)(2,10)(3,15)(5,19)(6,20)(7,18)(8,30)(9,31)(11,33)(12,34)(13,39)(14,40)(16,42)(17,41)(21,47)(22,48)(23,45)(24,49)(25,50)(26,51)(27,46)(28,54)(29,55)(32,56)(35,57)(36,58)(37,59)(38,60)(43,62)(44,61)(52,63)(53,64)

Polycyclic presentation:

2^10|a^2=h,b^2=hi,c^2=h,g^2=i,i^2=j,b^a=bg,c^b=cij,d^a=dj,d^b=df,e^d=eh,f^a=fh,f^e=fj,g^a=gi,g^b=gi,g^c=gj,g^d=ghj,g^f=gj,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250830, #72281944, #72307980
maximal normal subgroup of	#72250830, #72281944, #72307980
maximal quotient of	#72257010

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