#494072

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×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	C₂²×C₄⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,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, 401268);