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×#27991 |
| 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 | |
1566 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)(6,7)(8,10)(11,12,14,16)(13,15,17,18)(19,20)(21,27)(22,26)(23,30)(24,29)(25,31)(28,33)(32,34);(4,7)(8,9)(11,13)(12,15)(14,17)(16,18)(19,21,24,32,25,33,23,22)(20,26,30,28,31,34,29,27);(13,17)(15,18)(19,22,23,33,25,32,24,21)(20,27,29,34,31,28,30,26);(3,5,6,10)(4,8,7,9)(19,22,23,33,25,32,24,21)(20,28,29,26,31,27,30,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364988);