Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13244 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 70 points:
(1,2);(4,6)(7,8,13,20,15,21,14,39)(9,22,26,42,27,50,38,53)(10,28,31,60,33,61,32,47)(11,19,12,37,36,51,35,18)(16,40,43,24,45,25,44,34)(17,46,48,63,49,29,59,30)(23,41,56,66,57,68,64,67)(52,62,58,65,69,54,70,55);(3,4,5,6)(7,9,11,34,14,26,35,40,15,27,36,24,13,38,12,25)(8,16,37,22,20,44,51,53,21,45,18,50,39,43,19,42)(10,23,29,62,32,56,63,65,33,57,46,54,31,64,30,55)(17,41,60,52,48,67,61,70,49,68,47,69,59,66,28,58);(7,10,11,29,14,32,35,63,15,33,36,46,13,31,12,30)(8,17,37,60,20,48,51,61,21,49,18,47,39,59,19,28)(9,23,34,62,26,56,40,65,27,57,24,54,38,64,25,55)(16,41,22,52,44,67,53,70,45,68,50,69,43,66,42,58)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402425);