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×#55887 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4332 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)(5,9)(6,14)(7,8)(10,15)(11,12)(13,18)(16,17)(19,20,23,24)(21,27,29,25)(22,30,31,26)(28,33,32,34)(36,40)(37,43)(38,46)(48,50);(4,8)(5,11)(6,15)(16,18)(20,24)(22,31)(25,27)(28,32)(35,36,39,40)(37,44,43,41)(38,42,46,47)(45,50,49,48);(3,5,7,11)(4,9,8,12)(6,13,15,17)(10,16,14,18)(19,21)(20,25)(22,32)(23,29)(24,27)(26,34)(28,31)(30,33)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47)(49,50);(3,4)(5,12)(6,10)(7,8)(9,11)(13,18)(14,15)(16,17)(19,22,23,31)(20,26,24,30)(21,28,29,32)(25,33,27,34)(35,37,39,43)(36,41,40,44)(38,45,46,49)(42,48,47,50);(3,6)(4,10)(5,13)(7,15)(8,14)(9,16)(11,17)(12,18)(19,20)(21,25)(22,26)(23,24)(27,29)(28,33)(30,31)(32,34)(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, 8010034);