Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53330 |
| 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 | |
4230 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);(5,11)(9,14)(12,17)(16,18)(19,20,23,26)(21,24,28,31)(22,25,29,32)(27,30,33,34)(35,36)(37,43)(38,46)(39,42)(40,45)(41,47)(44,48)(49,50);(4,8)(6,13)(9,14)(12,17)(22,29)(25,32)(27,33)(30,34)(35,36,39,42)(37,43,45,40)(38,46,47,41)(44,48,50,49);(3,4,7,8)(5,9,11,14)(6,10,13,15)(12,16,17,18)(19,20)(21,24)(22,25)(23,26)(27,30)(28,31)(29,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,45)(44,48)(46,47)(49,50);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,21,23,28)(20,24,26,31)(22,27,29,33)(25,30,32,34)(35,37)(36,40)(38,44)(39,45)(41,48)(42,43)(46,49)(47,50);(3,6)(4,10)(5,12)(7,13)(8,15)(9,16)(11,17)(14,18)(19,22)(20,25)(21,27)(23,29)(24,30)(26,32)(28,33)(31,34)(35,38,39,47)(36,41,42,46)(37,44,45,50)(40,48,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013111);