#126329

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#31188	C₂
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

365 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

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

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, 33525);