#124287

order	512
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

353 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31483);