#497472

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

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×C₂²⋊C₈
prime radical	#3475	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

824 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404668);