#512837

order	512
nilpotency class	5
isoclinism	isoclinic class of #512832
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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 ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

742 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 146 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)(7,18)(8,14)(9,26)(10,32)(11,23)(12,17)(13,20)(15,29)(16,28)(19,46)(21,52)(22,48)(24,56)(25,44)(27,43)(30,62)(31,45)(33,41)(34,55)(35,47)(36,49)(37,57)(38,59)(39,51)(40,58)(42,50)(53,67)(54,61)(60,65)(63,66)(64,68);(5,7,12,22)(6,13,17,39)(8,19,11,35)(9,27,31,60)(10,33,34,64)(14,36,23,38)(15,41,42,68)(16,43,44,65)(18,45,48,26)(20,50,51,29)(21,25,53,28)(24,57,30,63)(32,37,55,66)(40,67,54,52)(46,62,47,56)(49,61,59,58);(2,4)(5,6)(7,13)(8,23)(9,15)(10,16)(11,14)(12,17)(18,20)(19,38)(21,37)(22,39)(24,54)(25,55)(26,29)(27,41)(28,32)(30,40)(31,42)(33,43)(34,44)(35,36)(45,50)(46,59)(47,49)(48,51)(52,57)(53,66)(56,61)(58,62)(60,68)(63,67)(64,65);(5,8,12,11)(6,14,17,23)(7,19,22,35)(9,24,31,30)(10,25,34,28)(13,36,39,38)(15,40,42,54)(16,32,44,55)(18,46,48,47)(20,49,51,59)(21,33,53,64)(26,56,45,62)(27,57,60,63)(29,58,50,61)(37,65,66,43)(41,67,68,52)

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,e^a=eg,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,d^c=di

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, 420033);