#72281744

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346546
commutator, Frattini, cosocle	C₂×#131	C₂⁴
prime radical	#10584275	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=h,c^2=hj,d^2=efgi,f^2=i,g^2=j,i^2=j,b^a=bh,c^b=ch,d^a=de,d^b=df,d^c=defgi,e^b=eh,f^a=fh,f^b=fi,f^c=fi,f^d=fj,f^e=fj,g^a=gh,g^b=gh,g^d=gj,g^e=gj,g^f=gj,h^d=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281944, #72307130, #72307980
maximal normal subgroup of	#72281944, #72307130, #72307980

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