#127160

order	512
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

429 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)(9,10,12,18,15,20,27,19,14,17,26,34,29,35,13,28)(11,16,22,31,24,33,23,39,21,30,25,36,38,40,37,32);(2,5)(3,7)(9,11,13,23,29,38,26,22,14,21,27,37,15,24,12,25)(10,16,19,32,35,40,18,36,17,30,28,39,20,33,34,31);(11,21)(16,30)(22,25)(23,37)(24,38)(31,36)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=h,f^2=gh,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, 34356);