Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25199 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,9,13,10,14,8,24)(5,15,17,11,18,28,25,29)(6,19,21,30,7,22,23,12)(16,27,33,34,20,26,32,31);(3,5,9,17,10,18,8,25)(4,11,13,28,14,29,24,15)(6,20,21,32,7,16,23,33)(12,27,19,34,30,26,22,31);(3,6,10,7)(4,12,14,30)(5,16,18,20)(8,21,9,23)(11,26,29,27)(13,22,24,19)(15,31,28,34)(17,32,25,33);(3,7,10,6)(4,12,14,30)(5,16,18,20)(8,21,9,23)(11,27,29,26)(13,19,24,22)(15,31,28,34)(17,33,25,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346094);