Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25438 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2678 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);(3,4,6,5)(7,8)(9,20)(10,15)(11,26)(12,18)(13,19)(14,22)(16,27)(17,24)(21,35)(23,30)(25,29)(28,33)(31,37)(32,36)(34,38);(3,5,6,4)(7,9,12,22,13,10,24,23)(8,14,17,20,19,30,18,15)(11,21,27,36,28,25,38,37)(16,29,26,32,34,35,33,31);(4,5)(7,10)(8,15)(9,13)(11,21)(12,23)(14,17)(16,32)(18,30)(19,20)(22,24)(25,28)(26,35)(27,36)(29,33)(31,34)(37,38);(7,11,24,38,13,28,12,27)(8,16,18,33,19,34,17,26)(9,21,23,37,10,25,22,36)(14,29,15,31,30,35,20,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351493);