#125958

order	512
nilpotency class	4
isoclinism	isoclinic class of #125958
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7755
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

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:

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

Polycyclic presentation:

2^9|a^2=f,c^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=df,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, 33154);