#146045

order	512
nilpotency class	4
isoclinism	isoclinic class of #146045
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10619
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 226 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53241);