#99543

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle	C₂³	#386
prime radical	C₂×#496	C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

525 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=g,e^2=gh,f^2=i,b^a=bd,c^a=ce,e^a=eg,c^b=ci,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,f^c=fh

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, 6739);