#98044

order	512
nilpotency class	3
isoclinism	isoclinic class of #97894
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	#2682	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

600 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=h,e^2=g,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5240);