#415789

order	512
nilpotency class	3
isoclinism	isoclinic class of #393929
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1470 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1470 conjugacy classes of subgroups — too many to draw. Only its 216 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gh,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322985);