#124635

order	512
nilpotency class	4
isoclinism	isoclinic class of #11836
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

343 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fghi,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cg,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, 31831);