#126943

order	512
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,e^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fi,c^b=ci,d^b=di,e^b=eg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34139);