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×#55855 |
| 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);(4,8)(5,11)(10,15)(13,17)(19,20)(21,25)(22,26)(23,24)(27,29)(28,33)(30,32)(31,34)(35,36,41,39)(37,44,38,42)(40,47,48,43)(45,49,46,50);(6,14)(10,15)(13,17)(16,18)(20,23)(21,27)(22,30)(33,34)(35,37,41,38)(36,42,39,44)(40,45,48,46)(43,49,47,50);(3,4,7,8)(5,12,11,9)(6,10,14,15)(13,18,17,16)(19,21,24,27)(20,25,23,29)(22,31,30,28)(26,34,32,33)(35,38,41,37)(36,42,39,44)(40,46,48,45)(43,49,47,50);(3,5,7,11)(4,9,8,12)(6,13,14,17)(10,16,15,18)(19,22,24,30)(20,26,23,32)(21,28,27,31)(25,33,29,34)(35,39,41,36)(37,42,38,44)(40,47,48,43)(45,49,46,50);(3,6)(4,10)(5,13)(7,14)(8,15)(9,16)(11,17)(12,18)(19,23,24,20)(21,29,27,25)(22,32,30,26)(28,34,31,33)(35,40,41,48)(36,43,39,47)(37,45,38,46)(42,49,44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008888);