Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#28063 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,14,16)(13,15,17,18)(19,20)(21,26)(22,31)(23,33)(24,28)(25,29)(27,32)(30,34);(3,5,6,10)(4,7,8,9)(11,13)(12,15)(14,17)(16,18)(19,21,23,29,24,30,32,22)(20,25,27,26,28,31,33,34);(4,8)(7,9)(13,17)(15,18);(4,8)(7,9)(19,22,32,30,24,29,23,21)(20,26,33,25,28,34,27,31)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364573);