#512944

order	512
nilpotency class	5
isoclinism	isoclinic class of #512941
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

432 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|b^2=eg,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gh,h^a=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420140);