Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141367 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1961 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #496 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 70 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)(5,19)(6,9)(8,17)(10,16)(11,25,15,29)(13,28,26,21)(18,30)(20,31,23,27)(24,32)(33,34)(35,45)(36,51,55,41)(37,57)(38,60)(39,43)(40,48)(42,58)(44,53)(46,68)(47,69,76,95)(49,72)(50,80)(52,84,87,94)(54,89,90,91)(56,67,88,83)(59,61)(62,92,93,75)(63,73)(64,78)(65,86,81,96)(66,85,82,74)(70,79)(71,77);(1,3,6,12)(2,7)(4,15,18,11)(5,13,16,26)(8,23)(9,22)(10,21)(14,29)(17,31,32,27)(19,28)(20,24)(25,30)(33,35,46,73,38,50,37,48,39,49,58,79,53,77,59,78)(34,40,61,80,44,63,42,45,43,64,57,71,60,70,68,72)(36,52,85,92,56,76,90,96,55,87,74,75,88,47,54,86)(41,65,91,95,67,93,82,84,51,81,89,69,83,62,66,94);(1,4,16,32,6,18,5,17)(2,8,19,30,9,24,10,14)(3,11,26,31,12,15,13,27)(7,20,28,29,22,23,21,25)(33,36,53,88,39,55,38,56)(34,41,44,67,43,51,60,83)(35,47,50,52,49,76,77,87)(37,54,46,74,59,90,58,85)(40,62,70,81,64,93,63,65)(42,66,57,91,68,82,61,89)(45,69,80,84,72,95,71,94)(48,75,79,86,78,92,73,96)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=egi,d^2=ghi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48568);