#496142

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

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	C₂³	C₂×#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 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)(3,8)(4,6)(5,7)(9,10,14,20,15,19,25,35,16,21,24,17,13,30,11,22)(12,18,28,34,29,33,37,40,26,32,38,31,27,39,23,36);(1,3,5,4)(2,6,7,8)(12,26)(18,32)(23,37)(27,29)(28,38)(31,34)(33,39)(36,40);(9,11,13,24,16,25,15,14)(10,17,19,22,21,20,30,35)(12,23,27,38,26,37,29,28)(18,31,33,36,32,34,39,40);(1,4,5,3)(2,6,7,8)(9,12,14,28,15,29,25,37,16,26,24,38,13,27,11,23)(10,18,20,34,19,33,35,40,21,32,17,31,30,39,22,36)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=eg,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, 403338);