Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24283 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4512 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)(9,21)(10,25)(11,15)(12,19)(13,18)(14,20)(16,27)(17,24)(22,36)(23,31)(26,35)(28,34)(29,33)(30,37)(32,38);(3,4)(5,6)(7,9,12,11,14,24,13,23)(8,15,18,21,20,31,19,17)(10,22,27,26,28,37,29,38)(16,30,25,35,33,36,34,32);(4,6)(7,10)(8,16)(9,22)(11,26)(12,27)(13,29)(14,28)(15,30)(17,32)(18,25)(19,34)(20,33)(21,35)(23,38)(24,37)(31,36);(7,11,12,24,14,23,13,9)(8,17,18,15,20,21,19,31)(10,26,27,37,28,38,29,22)(16,32,25,30,33,35,34,36)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349778);