#126308

order	512
nilpotency class	4
isoclinism	isoclinic class of #125636
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1954 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

275 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fi,d^2=f,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,h^a=hi,d^b=df,e^b=eg,d^c=dg,e^c=eh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33504);