Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#28244 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(7,8,14,20)(9,17,23,32)(10,24,28,36)(11,29,22,15)(12,19,13,18)(16,30,35,27)(21,33,25,31)(26,38,37,34);(3,4,5,6)(7,9,13,11,14,23,12,22)(8,15,19,32,20,29,18,17)(10,25,30,37,28,21,27,26)(16,33,36,34,35,31,24,38);(4,6)(7,10)(8,16)(9,21)(11,26)(12,27)(13,30)(14,28)(15,31)(17,34)(18,24)(19,36)(20,35)(22,37)(23,25)(29,33)(32,38);(7,11,13,23,14,22,12,9)(8,17,19,15,20,32,18,29)(10,26,30,25,28,37,27,21)(16,34,36,31,35,38,24,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372700);