Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25182 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3123 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)(9,14)(10,22)(11,13)(12,17)(15,23)(16,26)(18,29)(19,34)(20,27)(21,30)(24,31)(25,32)(28,33)(35,37)(36,38);(3,5,6,4)(8,13)(9,18)(11,26)(15,22)(16,17)(19,33)(20,21)(23,24)(27,29)(31,32)(34,37)(35,36);(4,5)(7,9,12,21)(8,14,17,30)(10,19,25,35)(11,20,26,18)(13,27,16,29)(15,28,31,38)(22,34,32,37)(23,33,24,36);(7,10)(8,15)(9,19)(11,24)(12,25)(13,22)(14,28)(16,32)(17,31)(18,33)(20,36)(21,35)(23,26)(27,34)(29,37)(30,38)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346041);