#126859

order	512
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

459 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=h,f^2=gh,g^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,e^b=ei

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, 34055);