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×#58975 |
| 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);(4,8)(5,11)(10,15)(13,17)(19,20,24,28)(21,25,29,32)(22,26,33,30)(23,27,34,31)(35,36)(37,39)(38,42)(40,48)(41,44)(43,49)(45,50)(46,47);(6,14)(10,15)(13,17)(16,18)(21,29)(23,34)(25,32)(27,31)(35,37,41,38)(36,42,44,39)(40,45,49,46)(43,50,48,47);(3,4,7,8)(5,12,11,9)(6,10,14,15)(13,18,17,16)(19,21,24,29)(20,25,28,32)(22,31,33,27)(23,30,34,26)(35,38,41,37)(36,42,44,39)(40,46,49,45)(43,50,48,47);(3,5,7,11)(4,9,8,12)(6,13,14,17)(10,16,15,18)(19,22)(20,26)(21,27)(23,32)(24,33)(25,34)(28,30)(29,31)(35,39,41,42)(36,37,44,38)(40,47,49,50)(43,45,48,46);(3,6,7,14)(4,10,8,15)(5,13,11,17)(9,16,12,18)(19,23,24,34)(20,27,28,31)(21,30,29,26)(22,32,33,25)(35,40)(36,43)(37,45)(38,46)(39,47)(41,49)(42,50)(44,48)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012433);