#126469

order	512
nilpotency class	4
isoclinism	isoclinic class of #125892
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1629
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#31071	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

388 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33665);