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×#29074 |
| 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×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1766 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)(5,9,10,8)(11,12,14,16)(13,17,18,15)(19,20)(21,25)(22,30)(23,26)(24,27)(28,32)(29,33)(31,34);(4,7)(8,9)(11,13,14,18)(12,15,16,17)(19,21,28,31,23,22,24,29)(20,25,32,34,26,30,27,33);(12,16)(15,17)(19,22,23,21)(20,25,26,30)(24,29,28,31)(27,34,32,33);(3,5,6,10)(4,8,7,9)(12,16)(15,17)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367767);