Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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×#366 |
| prime radical | C24 | Q32 |
| 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 | |
248 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 102 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)(5,6,9,15,12,17,11,33)(7,18,21,43,24,45,23,55)(8,25,27,56,29,58,28,59)(10,31,30,41,32,60,26,16)(13,34,35,51,37,50,61,20)(14,38,39,48,40,47,57,19)(22,53,52,65,54,68,49,44)(36,46,42,62,63,67,64,66);(5,7,19,46,10,22,27,34,9,21,38,62,30,52,29,51,12,24,48,67,32,54,28,50,11,23,47,66,26,49,8,20)(6,13,25,44,16,36,57,55,33,61,59,68,60,64,40,45,17,37,58,65,41,63,39,43,15,35,56,53,31,42,14,18);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=dg,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 | #72286849, #72286872, #72286879, … |
| maximal normal subgroup of | #72286849, #72286872, #72286879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59488);