#512943

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	#33607	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

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

normal subgroups only

This group has 436 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) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|b^2=egi,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, 420139);