Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C16 | C2×C42 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C4×D16 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
284 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6,10,18,14,21,13,39)(7,22,25,55,28,56,27,64)(8,16,31,20,12,38,33,53)(9,19,34,50,35,52,32,47)(11,17,36,49,30,48,37,51)(15,40,42,60,45,59,66,24)(23,54,57,68,29,65,58,43)(26,41,61,67,63,46,62,44);(2,4)(5,7,11,26,12,23,34,40,10,25,36,61,33,57,35,60,14,28,30,63,8,29,32,59,13,27,37,62,31,58,9,24)(6,15,19,43,20,44,51,64,39,66,47,65,16,46,48,56,21,45,52,68,53,67,49,55,18,42,50,54,38,41,17,22);(5,8,10,31,14,12,13,33)(6,16,18,20,21,38,39,53)(7,23,25,57,28,29,27,58)(9,30,34,37,35,11,32,36)(15,41,42,67,45,46,66,44)(17,47,49,19,48,50,51,52)(22,54,55,68,56,65,64,43)(24,26,40,61,60,63,59,62)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gh,e^2=i,f^2=g,g^2=hi,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gh,h^a=hi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286872, #72286915, #72286916, … |
| maximal normal subgroup of | #72286872, #72286915, #72286916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59160);