Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#95 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | C4×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,21,26,22,28,36,42,23,27,20,25,34,40,35,41)(19,29,30,39,32,43,44,48,31,24,33,37,45,38,46,47);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,24,27,29)(20,33)(21,30)(22,32)(23,31)(25,39,26,37)(28,38,40,43)(34,45)(35,46)(36,44)(41,48,42,47);(17,20,22,35,23,21,34,36)(18,25,28,41,27,26,40,42)(19,30,32,44,31,33,45,46)(24,37,38,47,29,39,43,48)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,e^2=h,f^2=g,h^2=i,b^a=bd,d^a=dg,f^a=fi,c^b=ci,d^b=di,e^b=eg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34240);