#128915

order	512
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#102
prime radical	C₂³	#416
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

310 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eh,d^2=gi,e^2=i,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,c^b=ci,d^b=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36111);