#126436

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

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

normal subgroups only

This group has 329 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fgi,c^2=fgi,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, 33632);