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×#60996 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3198 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6162 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)(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)(20,26)(22,30)(23,36)(24,31)(25,32)(27,39)(28,42)(29,41)(33,45)(34,48)(35,47)(37,40)(38,46)(43,50)(44,49);(5,12)(8,11)(13,18)(15,17)(19,22)(20,27)(21,30)(23,34)(24,38)(25,35)(26,39)(28,43)(29,44)(31,46)(32,47)(33,37)(36,48)(40,45)(41,49)(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, 10483037);