Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62196 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3521 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4,7,10)(5,11,12,8)(6,14,15,9)(13,16,18,17)(19,20)(21,31)(22,35)(23,30)(24,29)(25,28)(26,34)(27,36)(32,46)(33,41)(37,44)(38,43)(39,42)(40,47)(45,49)(48,50);(3,5)(4,8)(6,13)(7,12)(9,16)(10,11)(14,17)(15,18)(19,21)(20,26)(22,32)(23,38)(24,34)(25,33)(27,40)(28,43)(29,31)(30,41)(35,50)(36,49)(37,48)(39,47)(42,45)(44,46);(19,22)(20,27)(21,32)(23,36)(24,37)(25,39)(26,40)(28,35)(29,42)(30,44)(31,45)(33,47)(34,48)(38,49)(41,46)(43,50);(5,12)(8,11)(13,18)(16,17)(19,23,24,25)(20,28,29,30)(21,33,34,38)(22,36,37,39)(26,41,31,43)(27,35,42,44)(32,47,48,49)(40,46,45,50);(3,6,7,15)(4,9,10,14)(5,13,12,18)(8,16,11,17)(20,29)(21,34)(27,42)(28,30)(32,48)(33,38)(35,44)(47,49)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486643);