#512938

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂²	C₂²×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 440 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)(5,6)(7,18)(8,14)(9,27)(10,31)(11,33)(12,17)(13,22)(15,19)(16,28)(20,40)(21,41)(23,54)(24,55)(25,56)(26,39)(29,59)(30,58)(32,60)(34,48)(35,50)(36,63)(37,45)(38,47)(42,44)(43,52)(46,57)(49,68)(51,65)(53,61)(62,67)(64,66);(5,7,19,44,10,23,25,50,8,21,45,67,26,51,30,48,12,20,47,66,32,49,29,53,11,24,46,63,28,52,9,22)(6,13,27,43,16,36,57,55,33,61,59,68,60,64,38,40,17,34,58,65,39,62,37,41,14,35,56,54,31,42,15,18);(2,4)(7,20)(13,34)(18,40)(21,24)(22,48)(23,49)(35,61)(36,62)(41,55)(42,64)(43,65)(44,66)(50,53)(51,52)(54,68)(63,67);(5,8,12,11)(6,14,17,33)(7,21,20,24)(9,25,30,29)(10,26,32,28)(13,35,34,61)(15,37,38,57)(16,31,39,60)(18,41,40,55)(19,45,47,46)(22,50,48,53)(23,51,49,52)(27,56,58,59)(36,42,62,64)(43,54,65,68)(44,67,66,63)

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,c^b=ci

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, 420134);