#135008

order	512
nilpotency class	4
isoclinism	isoclinic class of #135007
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8648
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

281 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,d^2=g,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42204);