#145964

order	512
nilpotency class	4
isoclinism	isoclinic class of #145950
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2065
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,4)(5,7)(6,8)(9,10,14,17,13,23,21,24,15,12,22,19,16,18,11,20);(1,3,5,8,6,7,4,2)(9,11,16,22,15,21,13,14)(10,17,23,24,12,19,18,20);(2,3)(4,5)(7,8)(9,12)(10,15)(11,17)(13,23)(14,20)(16,18)(19,21)(22,24)

Polycyclic presentation:

2^9|a^2=deh,b^2=dg,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgi,c^b=cf,d^c=dgi,e^c=eh,f^c=fgi,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, 53160);