#145057

order	512
nilpotency class	4
isoclinism	isoclinic class of #143326
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1634
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10412
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

285 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=eg,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52253);