#509127

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×#2184	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

885 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^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, 416323);