#146292

order	512
nilpotency class	4
isoclinism	isoclinic class of #146292
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10690
commutator, Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

215 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,8,29,7,12,25,57)(3,14,30,43,18,41,56,34)(4,10,36,58,23,40,31,59)(5,11,28,13,15,42,6,26)(9,32,60,17,35,19,46,24)(16,45,51,37,49,64,54,62)(20,50,55,61,52,63,48,33)(21,53,27,39,47,38,22,44);(1,3,15,17)(2,9,11,41)(4,20,47,51)(5,24,7,18)(6,19,8,30)(10,37,53,50)(12,35,42,14)(13,43,29,46)(16,36,48,22)(21,54,23,52)(25,56,28,32)(26,34,57,60)(27,49,31,55)(33,59,45,44)(38,63,40,62)(39,61,58,64);(1,4,5,21,7,23,15,47)(2,10,11,38,12,40,42,53)(3,16,17,48,18,49,24,55)(6,27,25,36,28,22,8,31)(9,33,14,45,35,61,41,64)(13,44,29,58,26,39,57,59)(19,20,30,51,32,52,56,54)(34,37,46,50,43,62,60,63)

Polycyclic presentation:

2^9|a^2=def,b^2=de,c^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,f^a=fhi,g^a=gi,c^b=cf,d^b=dh,e^b=eh,f^b=fgi,h^b=hi,e^c=eg,f^c=fhi,g^c=gi,e^d=ei

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53488);