#136486

order	512
nilpotency class	4
isoclinism	isoclinic class of #136374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	Q₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

289 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,8,6,9,4,13)(3,10,11,7,12,15,14,16)(18,24,26,19)(20,30)(21,31)(22,32)(25,27,29,28);(1,3,4,14,6,12,5,11)(2,7,13,10,9,16,8,15)(17,18,22,28,23,26,32,27)(19,20,25,30,24,31,29,21);(17,19)(18,20)(21,27)(22,25)(23,24)(26,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=hi,f^2=ghi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43682);