#499070

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

1196 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)(6,7)(9,10,14,20)(11,24,27,12)(13,23,17,19)(15,35,16,25)(18,32,37,31)(21,33,22,34)(26,30,39,40)(28,38,29,36);(5,6)(7,8)(9,11)(10,12)(13,26)(14,27)(15,18)(16,37)(17,39)(19,30)(20,24)(21,36)(22,38)(23,40)(25,32)(28,34)(29,33)(31,35);(2,4)(5,7,8,6)(9,12,14,24)(10,18,20,37)(11,25,27,35)(13,30,17,40)(15,36,16,38)(19,29,23,28)(21,26,22,39)(31,33,32,34);(9,13,14,17)(10,19,20,23)(11,26,27,39)(12,30,24,40)(15,33,16,34)(18,29,37,28)(21,25,22,35)(31,36,32,38)

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=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, 406266);