#153251

order	512
nilpotency class	5
isoclinism	isoclinic class of #153083
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

261 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,e^c=egi,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, 60447);