Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13536 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,14,31,26,36,10,17,13,18,28,15,12,19,11,27)(9,20,29,33,30,35,21,34,24,32,25,16,23,38,22,37);(3,4)(5,6)(8,15,18,31)(10,14)(11,28)(12,26)(16,32,33,20)(17,19,27,36)(21,29)(22,25)(23,30)(34,38,37,35);(4,6)(7,9,13,24)(8,16,18,33)(10,25,11,29)(12,30,26,23)(14,22,28,21)(15,20,31,32)(17,35,27,38)(19,37,36,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406739);