Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#54551 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4,8,5)(6,9,13,11)(7,12,15,10)(14,17,18,16)(19,20)(21,26)(22,27)(23,31)(24,25)(28,32)(29,30)(33,34)(35,36,40,43)(37,38,45,41)(39,47,48,42)(44,49,50,46);(4,5)(6,13)(7,15)(16,17)(20,25)(22,26)(28,31)(30,33)(35,37,40,45)(36,41,43,38)(39,44,48,50)(42,49,47,46);(3,5,8,4)(6,9,13,11)(7,12,15,10)(14,16,18,17)(19,21,24,27)(20,26,25,22)(23,29,32,34)(28,33,31,30)(35,37)(36,41)(38,43)(39,44)(40,45)(42,49)(46,47)(48,50);(3,6,8,13)(4,9,5,11)(7,14,15,18)(10,16,12,17)(19,22,24,26)(20,27,25,21)(23,30,32,33)(28,34,31,29)(35,38,40,41)(36,37,43,45)(39,46,48,49)(42,44,47,50);(3,7)(4,10)(5,12)(6,14)(8,15)(9,16)(11,17)(13,18)(19,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,33)(27,34)(35,39,40,48)(36,42,43,47)(37,44,45,50)(38,46,41,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010117);