Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#33285 |
| 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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1718 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)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18)(19,20,24,27)(21,25,29,32)(22,31,26,23)(28,34,33,30);(4,8)(9,10)(11,13,14,17)(12,15,16,18)(19,21,24,29)(20,25,27,32)(22,28,26,33)(23,30,31,34);(3,5,7,6)(4,9,8,10)(19,22,23,20,24,26,31,27)(21,28,30,25,29,33,34,32);(3,6,7,5)(4,10,8,9)(12,16)(15,18)(21,29)(25,32)(28,33)(30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537872);