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×#54702 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4408 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);(4,8)(5,11)(10,15)(13,17)(19,20)(21,25)(22,30)(23,24)(26,31)(27,28)(29,34)(32,33)(35,36,40,41)(37,38,42,45)(39,47,48,43)(44,50,49,46);(6,14)(10,15)(13,17)(16,18)(20,24)(21,27)(22,31)(32,34)(36,41)(38,45)(43,47)(46,50);(3,4,7,8)(5,12,11,9)(6,10,14,15)(13,18,17,16)(19,21,23,27)(20,25,24,28)(22,29,31,33)(26,32,30,34)(35,37,40,42)(36,38,41,45)(39,44,48,49)(43,46,47,50);(3,5,7,11)(4,9,8,12)(6,13,14,17)(10,16,15,18)(19,20)(21,28)(22,26)(23,24)(25,27)(29,34)(30,31)(32,33)(35,38)(36,42)(37,41)(39,46)(40,45)(43,49)(44,47)(48,50);(3,6,7,14)(4,10,8,15)(5,13,11,17)(9,16,12,18)(19,22)(20,26)(21,29)(23,31)(24,30)(25,32)(27,33)(28,34)(35,39)(36,43)(37,44)(38,46)(40,48)(41,47)(42,49)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009947);