Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#60252 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
5966 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)(19,20,23,27,24,28,34,40)(21,25,31,37,30,36,43,47)(22,26,33,39,32,38,44,48)(29,35,42,46,41,45,49,50);(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)(26,35)(27,37)(28,36)(32,41)(33,42)(34,43)(38,45)(39,46)(40,47)(44,49)(48,50);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,22)(20,26)(21,29)(23,33)(24,32)(25,35)(27,39)(28,38)(30,41)(31,42)(34,44)(36,45)(37,46)(40,48)(43,49)(47,50);(4,10)(8,11)(9,18)(16,17)(22,32)(26,38)(29,41)(33,44)(35,45)(39,48)(42,49)(46,50);(21,30)(25,36)(29,41)(31,43)(35,45)(37,47)(42,49)(46,50)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481522);