#148573

order	512
nilpotency class	4
isoclinism	isoclinic class of #129405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1588 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1588 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,g^c=gh,g^e=gi

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, 55769);