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×#60876 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6028 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)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,25,29)(21,26,32,41)(22,34,36,42)(23,28,24,37)(27,38,43,35)(30,44,46,49)(31,40,33,47)(39,48,50,45);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21)(20,26)(22,30)(23,31)(24,33)(25,32)(27,39)(28,40)(29,41)(34,44)(35,45)(36,46)(37,47)(38,48)(42,49)(43,50);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,22)(20,27)(21,30)(23,35)(24,38)(25,36)(26,39)(28,42)(29,43)(31,45)(32,46)(33,48)(34,37)(40,49)(41,50)(44,47);(4,10)(8,11)(9,18)(16,17)(19,23,25,24)(20,28,29,37)(21,31,32,33)(22,35,36,38)(26,40,41,47)(27,42,43,34)(30,45,46,48)(39,49,50,44);(21,32)(26,41)(30,46)(31,33)(39,50)(40,47)(44,49)(45,48)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482914);