#98942

order	512
nilpotency class	3
isoclinism	isoclinic class of #98936
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

505 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6138);