#144498

order	512
nilpotency class	4
isoclinism	isoclinic class of #142220
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

657 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

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

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, 51694);