#126037

order	512
nilpotency class	4
isoclinism	isoclinic class of #125636
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

336 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,7)(3,5,8,6)(13,16)(17,18,21,25)(19,27,28,45)(20,31,32,46)(22,36,34,26)(23,38,39,29)(24,41,42,35)(30,47,37,43)(33,44,48,40);(2,5)(3,8)(6,7)(9,10,12,14)(11,13,15,16)(20,32)(24,42)(27,45)(29,38)(31,46)(33,48)(35,41)(43,47);(9,11)(10,13)(12,15)(14,16)(17,19,21,28)(18,23,25,39)(20,33,32,48)(22,37,34,30)(24,43,42,47)(26,44,36,40)(27,46,45,31)(29,41,38,35)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=i,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, 33233);