Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139160 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #466 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
273 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 273 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 96 points:
(1,2)(3,7,12,22)(4,14)(5,17)(6,9)(8,15)(10,19)(11,25,26,32)(13,29,28,21)(16,24)(18,23)(20,30,31,27)(33,34)(35,45,49,68)(36,41,52,66)(37,54)(38,59)(39,43)(40,48,63,55)(42,57)(44,60)(46,70)(47,75)(50,81,80,72)(51,82,56,85)(53,84,83,67)(58,61)(62,79)(64,86,88,73)(65,87,69,74)(71,78)(76,89)(77,92)(90,95)(91,93)(94,96);(1,3,6,12)(2,7)(4,11,16,26)(5,13,19,28)(8,20)(9,22)(10,21)(14,25)(15,27,18,30)(17,29)(23,31)(24,32)(33,35,46,73,60,80,58,48,39,49,57,86,38,50,37,55)(34,40,61,72,59,88,70,45,43,63,54,81,44,64,42,68)(36,47,74,93,83,96,51,79,52,78,87,95,53,77,56,76)(41,62,82,92,84,91,65,71,66,89,85,94,67,90,69,75);(1,4,6,16)(2,8,9,23)(3,11,12,26)(5,18,19,15)(7,20,22,31)(10,24,17,14)(13,30,28,27)(21,32,29,25)(33,36,38,53,39,52,60,83)(34,41,44,67,43,66,59,84)(35,47,50,77,49,78,80,96)(37,56,57,87,58,51,46,74)(40,62,64,90,63,89,88,91)(42,69,54,85,70,65,61,82)(45,71,72,92,68,75,81,94)(48,79,73,93,55,76,86,95)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=g,d^2=ghi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46373);