Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | C2×#357 |
| prime radical | #33600 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 98 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)(5,6,9,15,12,17,11,34)(7,18,21,44,24,46,23,57)(8,25,27,36,30,59,29,60)(10,32,31,42,33,61,26,16)(13,35,37,53,39,52,62,20)(14,19,40,48,41,49,58,28)(22,55,54,65,56,68,50,45)(38,51,43,47,63,66,64,67);(5,7,19,47,33,56,8,20,9,21,48,66,26,50,27,35,12,24,49,67,10,22,30,53,11,23,28,51,31,54,29,52)(6,13,36,55,42,63,41,46,34,62,25,45,32,43,40,44,17,39,60,68,16,38,14,18,15,37,59,65,61,64,58,57);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi140 conjugacy classes — view the full character table.
| maximal subgroup of | #72286872, #72320873, #72320903 |
| maximal normal subgroup of | #72286872, #72320873, #72320903 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59829);