Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19552 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19575 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1614 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);(3,4,8,11)(5,12,14,17)(6,10,7,16)(9,15,18,13)(19,20,24,28)(21,25,30,39)(22,32,34,42)(23,35,37,27)(26,36,41,33)(29,43,44,50)(31,47,46,40)(38,48,49,45);(3,5,7,13,8,14,6,15)(4,9,10,12,11,18,16,17)(19,21)(20,25)(22,29)(23,31)(24,30)(26,38)(27,40)(28,39)(32,43)(33,45)(34,44)(35,47)(36,48)(37,46)(41,49)(42,50);(3,4)(5,9)(6,16)(7,10)(8,11)(12,13)(14,18)(15,17)(19,22,24,34)(20,26,28,41)(21,29,30,44)(23,36,37,33)(25,38,39,49)(27,42,35,32)(31,48,46,45)(40,50,47,43);(3,6,8,7)(4,10,11,16)(5,13,14,15)(9,17,18,12)(21,30)(25,39)(29,44)(31,46)(38,49)(40,47)(43,50)(45,48)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261872);