#136365

order	512
nilpotency class	4
isoclinism	isoclinic class of #12366
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1629
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8886
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

329 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,d^b=dg,e^b=eh,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,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, 43561);