Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #121 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1797 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
294 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,6,9,7,4,5,15)(3,10,13,11,14,16,12,8)(17,18,20,24,21,25,30,36,22,26,31,37,32,38,42,46)(19,23,28,34,29,35,41,45,27,33,39,43,40,44,47,48);(1,3,6,13,7,14,5,12)(2,8,15,16,4,11,9,10)(17,19)(18,23)(20,28)(21,29)(22,27)(24,34)(25,35)(26,33)(30,41)(31,39)(32,40)(36,45)(37,43)(38,44)(42,47)(46,48);(1,4)(2,7)(3,11)(5,15)(6,9)(8,14)(10,13)(12,16)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,e^c=eg,f^c=fg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41920);