#124370

order	512
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 309 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=fg,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=di,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, 31566);