Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9327 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1334 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);(3,4,7,10,8,11,6,16)(5,12,15,17,13,18,14,9)(19,20)(21,24,28,32)(22,25)(23,26)(27,30)(29,33,34,31);(3,5)(4,9,11,17)(6,14)(7,15)(8,13)(10,12,16,18)(19,21)(20,24,26,32)(22,29)(23,28)(25,33,27,31)(30,34);(5,13)(9,17)(12,18)(14,15)(20,25)(21,28)(22,30)(24,31)(26,27)(32,33)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106625);