#126506

order	512
nilpotency class	4
isoclinism	isoclinic class of #126351
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

265 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=fi,d^2=f,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=dfi,e^b=egi,d^c=dg,e^c=ehi,h^c=hi,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33702);