#126646

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#95
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 ▷ C₂ ▷ 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)(9,10,12,18,15,20,29,36,14,17,28,33,13,19,27,35)(11,21,23,34,25,31,38,39,22,16,26,30,24,32,37,40);(1,2,4,7,5,8,3,6)(9,11,13,24,14,22,15,25)(10,16,19,31,17,21,20,32)(12,26,27,38,28,23,29,37)(18,34,35,40,33,30,36,39);(11,22)(16,21)(23,26)(24,25)(30,34)(31,32)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gi,e^2=h,f^2=g,h^2=i,b^a=bd,d^a=dg,f^a=fgi,c^b=ci,e^b=ei

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, 33842);