#122674

order	512
nilpotency class	4
isoclinism	isoclinic class of #122554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

512 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29870);