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×#56905 |
| 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 | |
4148 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)(36,40)(37,43)(38,46)(48,50);(4,6)(5,11)(7,15)(16,18)(20,25)(22,26)(28,31)(30,33)(35,36,39,40)(37,44,43,41)(38,42,46,47)(45,50,49,48);(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)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47)(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)(35,37,39,43)(36,41,40,44)(38,45,46,49)(42,48,47,50);(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)(36,42)(37,45)(39,46)(40,47)(41,48)(43,49)(44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006919);