Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60999 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6146 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)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(15,18)(19,20)(21,26)(22,33)(23,28)(24,37)(25,29)(27,34)(30,45)(31,40)(32,41)(35,43)(36,42)(38,44)(39,48)(46,49)(47,50);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17)(19,21,25,32)(20,26,29,41)(22,30,35,47)(23,36,24,31)(27,39,44,49)(28,42,37,40)(33,45,43,50)(34,48,38,46);(5,12)(8,11)(13,18)(15,17)(19,22,25,35)(20,27,29,44)(21,30,32,47)(23,34,24,38)(26,39,41,49)(28,43,37,33)(31,46,36,48)(40,45,42,50);(4,9)(8,11)(10,16)(15,17)(19,23,25,24)(20,28,29,37)(21,31,32,36)(22,34,35,38)(26,40,41,42)(27,43,44,33)(30,46,47,48)(39,45,49,50);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)(21,32)(26,41)(30,47)(31,36)(39,49)(40,42)(45,50)(46,48)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483040);