#148819

order	512
nilpotency class	4
isoclinism	isoclinic class of #13343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator	C₂²×D₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 968 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=d,f^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,f^c=fi,h^c=hi,f^e=fi

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, 56015);