#127758

order	512
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×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	C₂×#340
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

455 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gi,e^2=i,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,c^b=chi,d^b=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34954);