#145532

order	512
nilpotency class	4
isoclinism	isoclinic class of #143951
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
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

328 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fh,g^b=gi,e^c=egh,f^c=fgh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52728);