#153241

order	512
nilpotency class	5
isoclinism	isoclinic class of #153086
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#33610	C₂
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

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

normal subgroups only

This group has 269 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:

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

Polycyclic presentation:

2^9|c^2=d,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cf,f^b=fi,e^c=eg,f^c=fi,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, 60437);