Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#30140 |
| 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 | |
1832 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);(3,4,6,7)(5,8,9,10)(11,12,14,15)(13,17,18,16)(19,20)(21,26)(22,24)(23,32)(25,27)(28,29)(30,31)(33,34);(4,7)(8,10)(12,15)(16,17)(19,21,24,30)(20,26,22,31)(23,29,25,33)(27,34,32,28);(5,9)(8,10)(11,13)(12,16)(14,18)(15,17)(19,22,24,20)(21,28,30,34)(23,32,25,27)(26,33,31,29);(3,5,6,9)(4,8,7,10)(13,18)(16,17)(19,23,24,25)(20,27,22,32)(21,29,30,33)(26,34,31,28)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341403);