#125843

order	512
nilpotency class	4
isoclinism	isoclinic class of #125842
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7745
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

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

normal subgroups only

This group has 346 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,d^2=f,e^2=h,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=eh,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, 33039);