#144657

order	512
nilpotency class	4
isoclinism	isoclinic class of #144656
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10392
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

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

normal subgroups only

This group has 404 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:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=d,c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,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, 51853);