#509134

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

829 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)(9,10,16,31,11,18,28,17,14,21,13,20,15,22,24,33)(12,25,38,34,30,40,29,19,27,39,26,35,23,37,32,36);(5,6)(7,8)(10,17)(11,15)(12,26)(13,28)(16,24)(18,31)(19,35)(20,22)(21,33)(23,32)(27,38)(29,30)(34,36)(37,40);(2,4)(6,8)(9,11,14,15)(10,18,21,22)(12,23,27,30)(13,24,16,28)(17,20,33,31)(19,34,36,35)(25,37,39,40)(26,29,38,32);(2,4)(9,12,14,27)(10,19,21,36)(11,23,15,30)(13,29,16,32)(17,25,33,39)(18,34,22,35)(20,37,31,40)(24,38,28,26)

Polycyclic presentation:

2^9|a^2=ei,c^2=i,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=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, 416330);