Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24807 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2532 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);(7,8,14,18,13,20,12,19)(9,21,30,17,11,29,23,15)(10,24,26,16,28,34,27,33)(22,32,36,35,25,31,37,38);(3,4)(5,6)(7,9,12,23,13,11,14,30)(8,15,18,21,20,17,19,29)(10,22,26,36,28,25,27,37)(16,31,24,35,33,32,34,38);(4,6)(7,10)(8,16)(9,22)(11,25)(12,26)(13,28)(14,27)(15,31)(17,32)(18,24)(19,34)(20,33)(21,35)(23,36)(29,38)(30,37);(7,11,14,23,13,9,12,30)(8,17,19,21,20,15,18,29)(10,25,27,36,28,22,26,37)(16,32,34,35,33,31,24,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350241);