#501663

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

666 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408859);