Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13697 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C4×D16 |
| prime radical | C23 | #508 |
| LCS | G ▷ C2×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 | |
178 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 86 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)(5,6,10,16,12,18,33,17)(7,13,21,36,22,37,11,34)(8,23,27,54,29,56,47,55)(9,30,31,62,32,64,59,63)(14,38,40,60,42,67,41,26)(15,20,43,50,45,51,44,25)(19,46,49,68,24,52,28,61)(35,65,66,57,39,58,53,48);(2,4)(5,7,10,21,12,22,33,11)(6,13,16,36,18,37,17,34)(8,24,27,28,29,19,47,49)(9,20,31,50,32,51,59,25)(14,39,40,53,42,35,41,66)(15,30,43,62,45,64,44,63)(23,52,54,61,56,46,55,68)(26,57,38,58,60,48,67,65);(5,8,25,58,11,28,59,67,33,47,51,57,22,24,32,60,12,29,50,65,21,49,31,38,10,27,20,48,7,19,9,26)(6,14,30,46,13,35,43,54,16,40,62,68,36,66,45,56,18,42,64,52,37,39,44,55,17,41,63,61,34,53,15,23)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=eg,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci92 conjugacy classes — view the full character table.
| maximal subgroup of | #72286843, #72286872, #72320831, … |
| maximal normal subgroup of | #72286843, #72286872, #72320831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60052);