#495284

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

566 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)(3,6)(4,5)(7,8)(9,10,14,20,15,19,29,35,16,21,28,18,13,30,12,27)(11,17,25,34,26,33,37,40,23,32,38,31,24,39,22,36);(2,5)(6,8)(9,11,14,25,15,26,29,37,16,23,28,38,13,24,12,22)(10,17,20,34,19,33,35,40,21,32,18,31,30,39,27,36);(9,12,13,28,16,29,15,14)(10,18,19,27,21,20,30,35)(11,22,24,38,23,37,26,25)(17,31,33,36,32,34,39,40);(1,3,4,7)(2,6,5,8)(11,23)(17,32)(22,37)(24,26)(25,38)(31,34)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402480);