#126662

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,d^a=dg,f^a=fgi,c^b=ci,d^b=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, 33858);