#500858

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,13,19)(7,20,26,49)(8,15,23,52)(9,31,35,48)(10,36,38,67)(11,18,12,41)(14,42,47,24)(16,53,55,22)(17,37,56,21)(25,58,39,59)(27,54,61,64)(28,46,32,45)(29,63,62,57)(30,44,40,51)(33,43,34,66)(50,65,68,60);(5,7,21,32,12,25,38,42,13,26,37,28,11,39,10,24)(6,14,36,59,18,46,56,49,19,47,67,58,41,45,17,20)(8,27,60,33,30,22,62,9,23,61,65,34,40,53,29,35)(15,48,63,55,51,66,68,64,52,31,57,16,44,43,50,54);(2,4)(5,8)(6,15)(7,22)(9,32)(10,29)(11,40)(12,30)(13,23)(14,43)(16,20)(17,50)(18,51)(19,52)(21,60)(24,33)(25,61)(26,53)(27,39)(28,35)(31,45)(34,42)(36,63)(37,65)(38,62)(41,44)(46,48)(47,66)(49,55)(54,59)(56,68)(57,67)(58,64);(5,9,21,61,12,34,38,53,13,35,37,27,11,33,10,22)(6,16,36,43,18,54,56,48,19,55,67,66,41,64,17,31)(7,23,32,65,25,40,42,29,26,8,28,60,39,30,24,62)(14,44,59,50,46,15,49,63,47,51,58,68,45,52,20,57)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=eh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408054);