#124682

order	512
nilpotency class	4
isoclinism	isoclinic class of #124681
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1514
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

283 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31878);