#144638

order	512
nilpotency class	4
isoclinism	isoclinic class of #144637
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10390
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

424 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=i,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,h^b=hi,d^c=dg,f^c=fi,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, 51834);