#509145

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×#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

819 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)(9,10,13,18,29,36,39,33,14,19,30,34,15,20,28,17)(11,21,23,27,31,40,37,12,24,26,38,32,25,35,22,16);(5,6)(7,8)(9,11,14,24)(10,16,19,12)(13,22,30,37)(15,31,29,25)(17,21,33,26)(18,35,34,40)(20,27,36,32)(23,39,38,28);(2,4)(6,8)(9,12,14,16)(10,11,19,24)(13,26,30,21)(15,27,29,32)(17,22,33,37)(18,23,34,38)(20,25,36,31)(28,40,39,35);(2,4)(10,17)(11,22)(13,28)(15,29)(18,20)(19,33)(21,35)(23,25)(24,37)(26,40)(27,32)(30,39)(31,38)(34,36)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^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, 416341);