#72342834

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#9495
commutator	C₂²×D₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#110029
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=ef,c^2=gh,d^2=j,b^a=be,c^a=cfg,c^b=cfg,d^b=di,d^c=dh,e^a=ei,e^b=ei,e^c=eij,f^b=fi,f^c=fh,g^a=gh,g^b=ghj,g^e=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342838, #72342849, #72342855
maximal normal subgroup of	#72342838, #72342849, #72342855

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