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×#57771 |
| 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 | |
4284 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);(3,4,8,6)(5,9,11,13)(7,14,15,10)(12,18,17,16)(19,20,23,26)(21,27,28,24)(22,25,30,32)(29,33,34,31)(35,36,40,43)(37,38,45,41)(39,47,48,42)(44,49,50,46);(4,6)(5,11)(7,15)(16,18)(21,28)(24,27)(29,34)(31,33)(35,37,40,45)(36,41,43,38)(39,44,48,50)(42,49,47,46);(3,5)(4,9)(6,13)(7,12)(8,11)(10,16)(14,18)(15,17)(19,21,23,28)(20,24,26,27)(22,29,30,34)(25,31,32,33)(35,37)(36,41)(38,43)(39,44)(40,45)(42,49)(46,47)(48,50);(3,6,8,4)(5,9,11,13)(7,14,15,10)(12,16,17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,34)(35,38,40,41)(36,37,43,45)(39,46,48,49)(42,44,47,50);(3,7,8,15)(4,10,6,14)(5,12,11,17)(9,16,13,18)(19,22,23,30)(20,25,26,32)(21,29,28,34)(24,31,27,33)(35,39)(36,42)(37,44)(38,46)(40,48)(41,49)(43,47)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008759);