#72258402

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#9495
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
prime radical	#6437622	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

4455 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=j,c^2=j,f^2=j,b^a=bf,c^a=ch,c^b=ce,d^b=dj,e^a=eh,f^a=fi,f^b=fj,f^c=fh,g^b=gij,g^c=gj,g^e=gj,h^a=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258409
maximal normal subgroup of	#72258409
maximal quotient of	#72266221, #72266297

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