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×#23914 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6154 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)(9,13,18,28)(10,20,21,32)(11,23,26,15)(14,25,30,22)(17,33,34,37)(19,31,24,29)(27,36,38,35);(3,4)(5,6)(7,9,12,18)(8,13,16,28)(10,17,21,34)(11,24,26,19)(14,27,30,38)(15,31,23,29)(20,37,32,33)(22,36,25,35);(4,6)(7,10)(8,14)(9,17)(11,25)(12,21)(13,27)(15,32)(16,30)(18,34)(19,36)(20,23)(22,26)(24,35)(28,38)(29,37)(31,33);(9,18)(11,26)(13,28)(15,23)(17,34)(20,32)(22,25)(27,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344639);