#494115

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

598 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)(9,10,14,22,15,23,31,38,16,21,30,37,13,20,12,19)(11,24,27,39,28,17,25,35,29,18,32,33,26,34,40,36);(2,5)(6,8)(9,11)(10,17,21,34)(12,25)(13,26)(14,32)(15,28)(16,29)(18,20,24,23)(19,33,38,39)(22,36,37,35)(27,30)(31,40);(9,11)(10,18)(12,25)(13,26)(14,32)(15,28)(16,29)(17,20)(19,36)(21,24)(22,39)(23,34)(27,30)(31,40)(33,37)(35,38);(1,3,4,7)(2,6,5,8)(9,12,13,30,16,31,15,14)(10,19,20,37,21,38,23,22)(11,25,26,27,29,40,28,32)(17,33,24,35,34,39,18,36)

Polycyclic presentation:

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

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, 401311);