Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#39210 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(12,15)(16,18)(19,20)(21,27)(22,30)(23,24)(25,29)(26,31)(28,32)(33,34);(3,4,6,8)(5,9,10,7)(20,24)(25,27)(26,30)(32,33);(3,4)(5,7)(6,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,21,23,29)(20,25,24,27)(22,28,31,34)(26,32,30,33);(11,13,14,17)(12,16,15,18)(19,22)(20,26)(21,28)(23,31)(24,30)(25,32)(27,33)(29,34);(3,5,6,10)(4,7,8,9)(19,20)(21,25)(22,26)(23,24)(27,29)(28,32)(30,31)(33,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544135);