#145928

order	512
nilpotency class	4
isoclinism	isoclinic class of #145928
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10591
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

264 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=de,b^2=dg,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,f^b=fi,d^c=dg,e^c=eh,f^c=fg,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53124);