#144793

order	512
nilpotency class	4
isoclinism	isoclinic class of #144792
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	#135
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 337 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=dg,e^2=gi,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,f^b=fi,d^c=di,e^c=egi,f^c=fi,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, 51989);