#506382

order	512
nilpotency class	4
isoclinism	isoclinic class of #502469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

610 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 154 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)(7,15)(8,14)(9,30)(10,34)(11,17)(12,19)(13,21)(16,31)(18,35)(20,42)(22,46)(23,48)(24,56)(25,59)(26,62)(27,44)(28,50)(29,45)(32,40)(33,39)(36,49)(37,63)(38,52)(41,58)(43,60)(47,55)(51,64)(53,68)(54,67)(57,65)(61,66);(5,7,10,21,35,55,63,33,12,23,36,32,11,22,31,9)(6,13,37,48,17,30,34,47,19,40,16,15,18,39,49,46)(8,24,57,25,27,54,60,52,29,53,26,58,28,20,51,61)(14,41,43,56,50,38,65,42,45,59,64,68,44,66,62,67);(2,4)(5,8)(6,14)(7,20)(9,25)(10,26)(11,28)(12,29)(13,38)(15,42)(16,43)(17,50)(18,44)(19,45)(21,52)(22,53)(23,54)(24,55)(27,35)(30,59)(31,60)(32,61)(33,58)(34,62)(36,57)(37,64)(39,41)(40,66)(46,68)(47,56)(48,67)(49,65)(51,63);(5,9,31,22,11,32,36,23,12,33,63,55,35,21,10,7)(6,15,34,13,18,47,37,39,19,48,49,40,17,46,16,30)(8,25,60,53,28,61,57,54,29,58,51,24,27,52,26,20)(14,42,62,38,44,56,64,41,45,67,65,66,50,68,43,59)

Polycyclic presentation:

2^9|b^2=eghi,d^2=eh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413578);