#129180

order	512
nilpotency class	4
isoclinism	isoclinic class of #12077
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1776
prime radical	C₂³	#441
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

228 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=di,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=di,e^c=ei

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, 36376);