#145596

order	512
nilpotency class	4
isoclinism	isoclinic class of #145594
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 326 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=g,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fgh,h^b=hi,e^c=egh,f^c=fh,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52792);