Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#56551 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2670 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,7,8)(5,6,9,10)(11,12,14,15)(13,16,17,18)(19,20,23,26)(21,27,28,24)(22,25,30,32)(29,33,34,31);(4,8)(5,9)(12,15)(13,17)(21,28)(24,27)(29,34)(31,33);(11,13,14,17)(12,16,15,18)(19,21)(20,24)(22,29)(23,28)(25,31)(26,27)(30,34)(32,33);(3,5)(4,6)(7,9)(8,10)(11,13)(12,16)(14,17)(15,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,34);(3,6)(4,5)(7,10)(8,9)(11,12)(13,16)(14,15)(17,18)(19,22,23,30)(20,25,26,32)(21,29,28,34)(24,31,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006199);