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×#23917 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6078 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)(9,14)(10,22)(11,16)(12,27)(13,18)(15,25)(17,20)(19,32)(21,31)(23,38)(24,34)(26,30)(28,33)(29,36)(35,37);(3,4)(5,6)(7,9,13,21)(8,14,18,31)(10,19,24,35)(11,12,26,20)(15,29,33,38)(16,27,30,17)(22,37,34,32)(23,28,36,25);(4,6)(7,10)(8,15)(9,19)(11,23)(12,28)(13,24)(14,29)(16,32)(17,34)(18,33)(20,25)(21,35)(22,27)(26,36)(30,37)(31,38);(7,11,13,26)(8,16,18,30)(9,20,21,12)(10,23,24,36)(14,17,31,27)(15,32,33,37)(19,25,35,28)(22,29,34,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344635);