Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61875 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
5886 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)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,24,27)(21,25,30,39)(22,32,34,42)(23,35,37,28)(26,36,41,33)(29,43,44,50)(31,47,46,40)(38,48,49,45);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21)(20,25)(22,29)(23,31)(24,30)(26,38)(27,39)(28,40)(32,43)(33,45)(34,44)(35,47)(36,48)(37,46)(41,49)(42,50);(19,22,24,34)(20,26,27,41)(21,29,30,44)(23,36,37,33)(25,38,39,49)(28,42,35,32)(31,48,46,45)(40,50,47,43);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(20,27)(21,30)(26,41)(28,35)(29,44)(31,46)(32,42)(45,48);(4,10)(8,11)(9,18)(16,17)(21,30)(25,39)(29,44)(31,46)(38,49)(40,47)(43,50)(45,48)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485469);