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×#57011 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2456 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(12,15)(13,17)(19,20,24,23)(21,27,28,25)(22,30,31,26)(29,32,34,33);(6,10)(8,9)(11,12,14,15)(13,18,17,16)(21,28)(25,27)(29,34)(32,33);(4,5)(8,9)(11,12,14,15)(13,16,17,18)(19,21,24,28)(20,25,23,27)(22,29,31,34)(26,32,30,33);(3,5,7,4)(6,9,10,8)(11,12)(13,16)(14,15)(17,18)(19,22,24,31)(20,26,23,30)(21,29,28,34)(25,32,27,33);(3,6,7,10)(4,8,5,9)(11,13,14,17)(12,16,15,18)(19,23,24,20)(21,27,28,25)(22,30,31,26)(29,33,34,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007409);