#480077

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#525	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1080 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)(3,7,8,5)(4,6)(9,10,13,18,14,19,27,34,15,20,12,17,25,32,26,33)(11,16,23,30,24,31,37,40,22,29,21,28,35,38,36,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,28)(18,30)(19,31)(20,29)(25,35)(26,36)(27,37)(32,38)(33,39)(34,40);(9,12,14,26,15,13,25,27)(10,17,19,33,20,18,32,34)(11,21,24,36,22,23,35,37)(16,28,31,39,29,30,38,40);(11,22)(16,29)(21,23)(24,35)(28,30)(31,38)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=f,c^2=h,f^2=h,h^2=i,b^a=be,e^a=eg,d^b=di,f^b=fg

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, 387273);