#509152

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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×#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

806 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)(9,10,16,21,30,22,31,33,14,11,13,18,15,19,35,28)(12,23,39,36,32,40,29,17,26,24,25,38,27,37,34,20);(5,6)(7,8)(9,11,14,10)(12,24,26,23)(13,28,16,33)(15,19,30,22)(17,25,20,39)(18,31,21,35)(27,37,32,40)(29,36,34,38);(2,4)(6,8)(10,11)(17,20)(18,21)(19,22)(23,24)(28,33)(36,38)(37,40);(2,4)(9,12,14,26)(10,17,11,20)(13,29,16,34)(15,32,30,27)(18,37,21,40)(19,38,22,36)(23,28,24,33)(25,35,39,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416348);