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×#61922 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6010 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)(5,13)(6,10)(7,11)(8,12)(9,15)(14,18)(16,17)(19,20)(21,26)(22,33)(23,28)(24,37)(25,29)(27,34)(30,45)(31,40)(32,42)(35,43)(36,41)(38,44)(39,47)(46,49)(48,50);(3,5,8,15)(4,9,12,13)(6,14,7,16)(10,17,11,18)(19,21,25,32)(20,26,29,42)(22,30,35,48)(23,36,24,31)(27,39,44,49)(28,41,37,40)(33,45,43,50)(34,47,38,46);(3,6,8,7)(4,10,12,11)(5,14,15,16)(9,17,13,18)(19,22)(20,27)(21,30)(23,34)(24,38)(25,35)(26,39)(28,43)(29,44)(31,46)(32,48)(33,37)(36,47)(40,45)(41,50)(42,49);(3,7,8,6)(4,11,12,10)(5,14,15,16)(9,17,13,18);(4,12)(9,13)(10,11)(17,18)(19,23,25,24)(20,28,29,37)(21,31,32,36)(22,34,35,38)(26,40,42,41)(27,43,44,33)(30,46,48,47)(39,45,49,50)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485795);