#494073

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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 ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

700 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)(3,6)(4,5)(7,8)(9,10,15,21,13,20,27,35,16,22,29,36,14,19,28,34)(11,18,25,33,12,17,26,32,23,31,37,40,24,30,38,39);(2,5)(6,8)(9,11,14,24,16,23,13,12)(10,17,20,31,22,30,19,18)(15,25,28,38,29,37,27,26)(21,32,35,40,36,39,34,33);(1,3,4,7)(2,6,5,8)(9,12,13,23,16,24,14,11)(10,18,19,30,22,31,20,17)(15,26,27,37,29,38,28,25)(21,33,34,39,36,40,35,32);(11,23)(12,24)(17,30)(18,31)(25,37)(26,38)(32,39)(33,40)

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,d^c=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, 401269);