Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#57757 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4,7,10)(5,11,12,8)(6,9,14,16)(13,17,18,15)(19,20,24,25)(21,22,26,29)(23,31,32,27)(28,34,33,30)(35,36)(37,40)(38,46)(39,42)(41,47)(43,45)(44,49)(48,50);(5,12)(8,11)(13,18)(15,17)(20,25)(22,29)(27,31)(30,34)(35,36,39,42)(37,43,45,40)(38,46,47,41)(44,48,50,49);(3,5,7,12)(4,8,10,11)(6,13,14,18)(9,15,16,17)(19,21,24,26)(20,22,25,29)(23,28,32,33)(27,30,31,34)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,47)(48,50);(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,26)(21,25)(23,30)(24,29)(27,33)(28,31)(32,34)(35,37)(36,40)(38,44)(39,45)(41,48)(42,43)(46,49)(47,50);(3,6,7,14)(4,9,10,16)(5,13,12,18)(8,15,11,17)(19,23)(20,27)(21,28)(22,30)(24,32)(25,31)(26,33)(29,34)(35,38)(36,41)(37,44)(39,47)(40,48)(42,46)(43,49)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009105);