#126566

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	#97
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

492 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cg,d^a=di,f^a=fi

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33762);