#153603

order	512
nilpotency class	6
isoclinism	isoclinic class of #153601
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₃₂
commutator, Frattini, cosocle	C₂×C₃₂	C₂³
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

183 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 44 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=df,f^b=fg,g^b=gh,h^b=hi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60799);