#145954

order	512
nilpotency class	4
isoclinism	isoclinic class of #145954
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10595
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:

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

Polycyclic presentation:

2^9|a^2=deh,b^2=d,d^2=g,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,d^c=dgi,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, 53150);