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×#54232 |
| 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 | |
4132 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)(5,8)(6,11)(7,12)(9,13)(10,15)(14,16)(17,18)(19,20,23,24)(21,27,29,25)(22,30,31,26)(28,32,34,33)(35,36)(37,39)(38,42)(40,48)(41,44)(43,49)(45,50)(46,47);(4,5)(6,13)(10,12)(14,18)(20,24)(22,31)(25,27)(28,34)(35,37,41,38)(36,42,44,39)(40,45,49,46)(43,50,48,47);(3,5,8,4)(6,11,13,9)(7,12,15,10)(14,17,18,16)(19,21,23,29)(20,25,24,27)(22,28,31,34)(26,32,30,33)(35,38,41,37)(36,42,44,39)(40,46,49,45)(43,50,48,47);(3,6)(4,9)(5,11)(7,14)(8,13)(10,16)(12,17)(15,18)(19,20)(21,27)(22,26)(23,24)(25,29)(28,33)(30,31)(32,34)(35,39,41,42)(36,37,44,38)(40,47,49,50)(43,45,48,46);(3,7,8,15)(4,10,5,12)(6,14,13,18)(9,16,11,17)(19,22,23,31)(20,26,24,30)(21,28,29,34)(25,32,27,33)(35,40,41,49)(36,43,44,48)(37,45,38,46)(39,47,42,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007012);