#499174

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,4)(9,10)(11,25)(12,31)(13,22)(14,21)(15,35)(16,36)(17,20)(18,27)(19,29)(23,32)(24,26)(28,39)(30,40)(33,38)(34,37);(5,6)(7,8)(9,11,17,30)(10,18,20,37)(12,32,28,13)(14,27,15,34)(16,33,26,29)(19,21,38,35)(22,25,23,40)(24,31,36,39);(2,4)(6,8)(9,12,13,11,26,27,15,33,17,28,32,30,16,34,14,29)(10,19,21,37,36,40,23,39,20,38,35,18,24,25,22,31);(10,20)(18,37)(19,38)(21,35)(22,23)(24,36)(25,40)(31,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406370);