#126254

order	512
nilpotency class	4
isoclinism	isoclinic class of #126253
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7782
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	#532	D₄
LCS	G ▷ C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33450);