Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32653 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,9)(5,6,10,8)(11,12)(13,17)(14,16)(15,18)(19,20)(21,29)(22,27)(23,25)(24,28)(26,31)(30,33)(32,34);(3,5,7,10)(4,8,9,6)(11,13)(12,15)(14,18)(16,17)(20,25)(21,30)(22,24)(26,33)(29,34)(31,32);(12,16)(15,17)(19,21,23,31)(20,26,25,29)(22,30,24,32)(27,34,28,33);(3,6,7,8)(4,5,9,10)(13,18)(15,17)(19,22,23,24)(20,27,25,28)(21,30,31,32)(26,34,29,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7418319);