#494104

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 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)(3,5)(4,6)(7,8)(9,10,14,19,13,18,27,35,15,20,29,37,12,16,26,31)(11,21,25,38,24,33,39,36,22,30,28,40,23,17,32,34);(1,3,4,7)(2,5,6,8)(10,16,20,18)(17,30,33,21)(19,31,37,35)(34,40,36,38);(2,6)(5,8)(9,11,12,23,15,22,13,24)(10,17,18,21,20,33,16,30)(14,28,26,39,29,25,27,32)(19,36,35,40,37,34,31,38);(11,22)(17,33)(21,30)(23,24)(25,28)(32,39)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,d^c=di,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, 401300);