#145953

order	512
nilpotency class	4
isoclinism	isoclinic class of #144059
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 243 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6,13,40,35,18,11,38,12,17,34,16,10,36,9,32)(7,19,50,43,22,52,21,44,25,53,41,65,24,60,23,14)(8,26,33,46,37,49,30,64,31,15,39,62,29,47,28,48)(20,51,59,42,55,45,54,66,57,68,58,67,56,61,27,63);(1,2)(3,4)(5,7,9,21,10,24,34,50,12,25,11,23,35,22,13,41)(6,14,16,19,18,43,32,52,17,44,40,53,36,65,38,60)(8,27,28,55,29,58,39,20,31,54,30,56,37,59,33,57)(15,45,46,66,47,68,64,67,26,61,62,63,49,51,48,42);(2,4)(5,8,12,31)(6,15,17,26)(7,20,25,57)(9,33,11,39)(10,37,35,29)(13,28,34,30)(14,42,44,67)(16,48,40,64)(18,49,36,47)(19,51,53,68)(21,58,23,59)(22,56,24,55)(27,41,54,50)(32,62,38,46)(43,63,65,66)(45,52,61,60)

Polycyclic presentation:

2^9|a^2=deh,b^2=d,c^2=i,d^2=g,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,e^b=ei,d^c=dgi,e^c=eh,f^c=fgi,g^c=gi

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, 53149);