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×#56975 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4186 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);(3,4,8,6)(5,9,11,13)(7,14,15,10)(12,18,17,16)(19,20)(21,26)(22,27)(23,31)(24,25)(28,32)(29,30)(33,34)(35,36,39,42)(37,43,45,40)(38,41,46,48)(44,49,50,47);(4,6)(5,11)(7,15)(16,18)(20,25)(22,26)(28,31)(30,33)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,5)(4,9)(6,13)(7,12)(8,11)(10,16)(14,18)(15,17)(19,21,24,27)(20,26,25,22)(23,29,32,34)(28,33,31,30)(35,36)(37,40)(38,41)(39,42)(43,45)(44,47)(46,48)(49,50);(3,6,8,4)(5,9,11,13)(7,14,15,10)(12,16,17,18)(19,22,24,26)(20,27,25,21)(23,30,32,33)(28,34,31,29)(35,37)(36,40)(38,44)(39,45)(41,47)(42,43)(46,50)(48,49);(3,7,8,15)(4,10,6,14)(5,12,11,17)(9,16,13,18)(19,23,24,32)(20,28,25,31)(21,29,27,34)(22,30,26,33)(35,38,39,46)(36,41,42,48)(37,44,45,50)(40,47,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006888);