Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10523 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,8,12)(5,13,10,6)(7,14,17,11)(9,16,18,15)(19,20)(21,29)(22,27)(23,26)(24,28)(25,31)(30,34)(32,33);(3,5,8,10)(4,9,12,18)(6,14,13,11)(7,16,17,15)(19,21)(20,25)(22,31)(23,32)(24,30)(26,34)(27,29)(28,33);(3,6,7,9,8,13,17,18)(4,10,11,16,12,5,14,15)(20,26)(21,30)(23,24)(27,28)(31,32)(33,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146065);