#144791

order	512
nilpotency class	4
isoclinism	isoclinic class of #144790
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10397
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

333 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51987);