#468347

order	512
nilpotency class	3
isoclinism	isoclinic class of #454045
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1848 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1848 conjugacy classes of subgroups — too many to draw. Only its 210 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 — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^c=hi,f^d=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, 375543);