Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13412 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,5)(7,8)(9,14)(10,15)(11,28)(12,29)(13,18)(16,25)(17,26)(19,37)(20,33)(21,38)(22,31)(23,35)(24,32)(27,36)(30,34);(3,4)(5,6)(7,9,13,23)(8,14,18,35)(10,24,27,21)(11,19,25,20)(12,30,26,22)(15,32,36,38)(16,33,28,37)(17,31,29,34);(3,5,6,4)(8,15)(9,19)(10,25)(11,27)(12,26)(14,31)(16,17)(18,36)(20,23)(21,30)(22,24)(28,29)(33,37)(34,35)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406280);