#126340

order	512
nilpotency class	4
isoclinism	isoclinic class of #125647
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1619
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=fi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,h^a=hi,d^b=df,e^b=eg,f^b=fi,d^c=dg,e^c=eh,h^c=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33536);