#513186

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
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 ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

430 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 120 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)(6,12)(7,17)(8,23)(9,26)(10,31)(13,36)(14,41)(15,45)(16,48)(18,39)(19,42)(20,34)(21,56)(22,30)(24,29)(25,51)(27,63)(28,35)(32,46)(33,38)(37,66)(40,64)(43,67)(44,53)(47,50)(49,57)(52,65)(55,68)(58,59)(61,62);(5,6,11,13)(7,18,22,41)(8,24,28,52)(9,29,32,65)(10,33,31,16)(12,35,36,23)(14,42,39,19)(15,46,40,26)(17,50,30,37)(20,55,56,67)(21,57,34,53)(25,47,62,66)(27,64,58,45)(38,59,48,63)(43,51,68,61)(44,54,49,60);(2,4)(6,13)(12,36)(14,39)(15,40)(16,33)(18,41)(24,52)(29,65)(37,50)(38,48)(43,68)(44,49)(45,64)(47,66)(53,57)(55,67);(5,7,11,22)(6,14,13,39)(8,25,28,62)(9,30,32,17)(10,34,31,21)(12,37,36,50)(15,47,40,66)(16,49,33,44)(18,52,41,24)(19,35,42,23)(20,46,56,26)(27,51,58,61)(29,43,65,68)(38,67,48,55)(45,57,64,53)(54,59,60,63)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420382);