Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24229 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3497 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4552 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);(4,5)(7,8)(9,13)(10,20)(11,23)(12,16)(14,22)(15,26)(17,34)(18,32)(19,29)(21,33)(24,28)(25,31)(27,36)(30,37)(35,38);(3,4,6,5)(7,9,12,19)(8,13,16,29)(10,17,21,35)(11,24,26,18)(14,30,31,27)(15,32,23,28)(20,34,33,38)(22,37,25,36);(7,10,12,21)(8,14,16,31)(9,17,19,35)(11,25,26,22)(13,27,29,30)(15,33,23,20)(18,37,24,36)(28,38,32,34);(3,5,6,4)(10,21)(14,31)(17,35)(20,33)(22,25)(27,30)(34,38)(36,37)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349755);