#134680

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#335
prime radical	C₂⁴	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci

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, 41876);