#512947

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
prime radical	#33610	C₂
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

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

normal subgroups only

This group has 426 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)(5,6,10,16)(7,19,22,48)(8,26,27,14)(9,31,33,65)(11,37,35,67)(12,38,36,18)(13,39,41,23)(15,44,45,20)(17,40,46,32)(21,53,54,47)(24,59,56,63)(25,60,42,51)(28,57,50,64)(29,58,62,68)(30,43,49,61)(34,66,52,55);(5,7,20,43,11,24,52,68,36,42,47,14,40,64,33,39,10,22,44,61,35,56,34,58,12,25,53,26,32,57,9,23)(6,13,31,50,17,27,54,60,38,62,66,63,67,30,45,48,16,41,65,28,46,8,21,51,18,29,55,59,37,49,15,19);(2,4)(5,8)(6,14)(7,21)(9,28)(10,27)(11,29)(12,30)(13,40)(15,42)(16,26)(17,39)(18,43)(19,47)(20,51)(22,54)(23,46)(24,55)(25,45)(31,64)(32,41)(33,50)(34,63)(35,62)(36,49)(37,68)(38,61)(44,60)(48,53)(52,59)(56,66)(57,65)(58,67);(7,22)(8,27)(13,41)(14,26)(19,48)(23,39)(24,56)(25,42)(28,50)(29,62)(30,49)(43,61)(51,60)(57,64)(58,68)(59,63)

Polycyclic presentation:

2^9|a^2=i,b^2=eg,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^b=di,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, 420143);