#146167

order	512
nilpotency class	4
isoclinism	isoclinic class of #146167
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10653
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#471
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

187 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53363);