#105208

order	512
nilpotency class	3
isoclinism	isoclinic class of #105185
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#396
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

299 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,3,8,5,4,6,7)(10,15)(11,16)(12,13)(18,21)(22,24);(1,3,5,6)(2,7,4,8)(9,10,13,11,14,16,12,15)(18,22)(20,23)(21,24);(1,4,5,2)(3,8,6,7)(9,11,14,15)(10,12,16,13)(17,18,19,21)(20,24,23,22)

Polycyclic presentation:

2^9|a^2=dei,b^2=dg,c^2=hi,d^2=g,e^2=h,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,f^b=fi,d^c=dg,e^c=eh,f^c=fgi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12404);