Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33806 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20)(21,31)(22,30)(23,26)(24,37)(25,28)(27,34)(29,39)(32,48)(33,47)(35,42)(36,44)(38,46)(40,45)(41,49)(43,50);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(20,26)(27,41)(28,30)(29,42)(31,47)(37,44)(43,46)(45,48);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,21,23,33)(20,27,26,41)(22,32,25,40)(24,38,36,50)(28,43,30,46)(29,45,42,48)(31,44,47,37)(34,35,49,39);(4,10)(8,11)(9,18)(16,17)(19,22,23,25)(20,28,26,30)(21,32,33,40)(24,35,36,39)(27,43,41,46)(29,44,42,37)(31,45,47,48)(34,38,49,50)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530604);