Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#45878 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5110 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);(4,6)(5,7)(12,15)(16,18)(20,24)(21,27)(26,31)(28,33);(3,4,8,6)(5,10,7,9)(20,24)(21,27)(22,29)(32,34);(3,5,8,7)(4,9,6,10)(11,12,14,15)(13,16,17,18)(19,20)(21,25)(22,26)(23,24)(27,30)(28,32)(29,31)(33,34);(3,6,8,4)(5,10,7,9)(11,13,14,17)(12,16,15,18)(19,21,23,27)(20,25,24,30)(22,28,29,33)(26,32,31,34);(3,7,8,5)(4,10,6,9)(19,22)(20,26)(21,28)(23,29)(24,31)(25,32)(27,33)(30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609631);