#146355

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂×C₄	C₄³
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1168 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 250 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=i,b^a=bd,d^a=dh,f^a=fi,h^a=hi,c^b=ci,e^b=ehi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53551);