#146134

order	512
nilpotency class	4
isoclinism	isoclinic class of #146134
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2077
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 206 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) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=eh,c^2=dh,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fg,c^b=cf,d^b=dg,e^b=egi,d^c=di,e^c=eh,f^c=fg,h^c=hi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53330);