#127946

order	512
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

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

464 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=hi,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, 35142);