#502059

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

813 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)(5,6,8,7)(9,10,14,18)(11,15,21,29)(12,16,25,33)(13,17,24,32)(19,27,36,39)(20,28,26,34)(22,30,35,38)(23,31,37,40);(2,4)(5,7,8,6)(11,19)(12,22)(13,24)(15,27)(16,30)(17,32)(20,37)(21,36)(23,26)(25,35)(28,40)(29,39)(31,34)(33,38);(6,7)(9,11,14,21)(10,15,18,29)(12,23,25,37)(13,26,24,20)(16,31,33,40)(17,34,32,28)(19,35,36,22)(27,38,39,30);(2,4)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,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, 409255);