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×#54445 |
| 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 | |
4164 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)(5,9,13,11)(7,14,15,10)(12,17,18,16)(19,20)(21,25)(22,26)(23,24)(27,29)(28,33)(30,32)(31,34)(35,36,39,42)(37,43,45,40)(38,41,46,48)(44,49,50,47);(4,6)(9,11)(10,14)(16,17)(20,23)(21,27)(22,30)(33,34)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,5,8,13)(4,9,6,11)(7,12,15,18)(10,16,14,17)(19,21,24,27)(20,25,23,29)(22,31,30,28)(26,34,32,33)(35,36)(37,40)(38,41)(39,42)(43,45)(44,47)(46,48)(49,50);(3,6,8,4)(5,11,13,9)(7,14,15,10)(12,17,18,16)(19,22,24,30)(20,26,23,32)(21,28,27,31)(25,33,29,34)(35,37)(36,40)(38,44)(39,45)(41,47)(42,43)(46,50)(48,49);(3,7,8,15)(4,10,6,14)(5,12,13,18)(9,16,11,17)(19,23,24,20)(21,29,27,25)(22,32,30,26)(28,34,31,33)(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, 8010813);