#126252

order	512
nilpotency class	4
isoclinism	isoclinic class of #125887
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1659
commutator, Frattini, cosocle	#376	C₂³
socle	C₂	#7782
prime radical	#532	D₄
LCS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #376 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 290 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:

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

Polycyclic presentation:

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

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, 33448);