#123263

order	512
nilpotency class	4
isoclinism	isoclinic class of #123243
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

348 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 42 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)(4,7)(6,8)(9,10,15,20)(11,21,25,39)(12,26,29,36)(13,17,14,19)(16,31,34,22)(18,30,37,28)(23,35,24,38)(27,32,40,33);(2,5)(6,7)(9,11,15,25)(10,16,20,34)(12,27,29,40)(13,24,14,23)(17,32,19,33)(18,35,37,38)(21,36,39,26)(22,30,31,28);(1,3,4,8)(2,6)(5,7)(10,17,20,19)(16,32,34,33)(18,36,37,26)(21,35,39,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=ehi,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30459);