#136679

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#447
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8947
prime radical	#3414	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

359 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43875);