#138860

order	512
nilpotency class	4
isoclinism	isoclinic class of #138832
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

383 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,d^2=ghi,e^2=ghi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egh,f^b=fh,h^b=hi,d^c=dgh,e^c=egh,f^c=fgh,h^c=hi

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, 46056);