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×#54563 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 512 |
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)(19,20,24,28)(21,25,29,34)(22,26,32,31)(23,27,33,30)(36,37)(38,45)(39,47)(48,49);(4,6)(5,11)(7,15)(16,18)(21,29)(23,33)(25,34)(27,30)(35,36,40,37)(38,43,45,41)(39,44,47,42)(46,48,50,49);(3,5)(4,9)(6,13)(7,12)(8,11)(10,16)(14,18)(15,17)(19,21)(20,25)(22,27)(23,31)(24,29)(26,33)(28,34)(30,32)(35,37,40,36)(38,41,45,43)(39,44,47,42)(46,48,50,49);(3,6,8,4)(5,9,11,13)(7,14,15,10)(12,16,17,18)(19,22)(20,26)(21,30)(23,25)(24,32)(27,29)(28,31)(33,34)(35,38,40,45)(36,41,37,43)(39,46,47,50)(42,48,44,49);(3,7,8,15)(4,10,6,14)(5,12,11,17)(9,16,13,18)(19,23)(20,27)(21,31)(22,25)(24,33)(26,29)(28,30)(32,34)(35,39)(36,42)(37,44)(38,46)(40,47)(41,48)(43,49)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010836);