Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32660 |
| 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 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1824 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);(3,4,7,9)(5,6,10,8)(11,12)(13,17)(14,16)(15,18)(19,20)(21,24)(22,26)(23,25)(27,31)(28,32)(29,33)(30,34);(3,5,7,10)(4,8,9,6)(11,13)(12,15)(14,18)(16,17)(21,27)(22,29)(24,31)(26,33)(28,30)(32,34);(12,16)(15,17)(19,21,23,28)(20,24,25,32)(22,30,29,27)(26,34,33,31);(3,6,7,8)(4,5,9,10)(13,18)(15,17)(20,25)(24,32)(26,33)(31,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7418325);