Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C2×#23916 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3487 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(7,8,12,16)(9,17,19,31)(10,14,21,29)(11,23,24,15)(13,25,28,20)(18,34,35,33)(22,32,26,30)(27,37,38,36);(3,4)(5,6)(10,21)(11,24)(14,29)(15,23)(17,31)(18,35)(20,25)(27,38);(4,6)(7,9)(8,13)(10,18)(11,25)(12,19)(14,27)(15,31)(16,28)(17,23)(20,24)(21,35)(22,37)(26,36)(29,38)(30,33)(32,34);(7,10,12,21)(8,14,16,29)(9,18,19,35)(11,26,24,22)(13,27,28,38)(15,32,23,30)(17,33,31,34)(20,37,25,36)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344637);