#153404

order	512
nilpotency class	5
isoclinism	isoclinic class of #153213
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2226
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

217 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=dh,b^2=deg,d^2=ghi,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=dg,e^c=eg,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60600);