Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#32601 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,9)(5,8,6,10)(12,15)(13,17)(19,20,22,25)(21,24,28,32)(23,26,30,33)(27,34,29,31);(3,4)(5,8)(6,10)(7,9)(11,12)(13,16)(14,15)(17,18)(19,21,22,28)(20,24,25,32)(23,27,30,29)(26,34,33,31);(3,5,7,6)(4,8,9,10)(11,13,14,17)(12,16,15,18)(20,25)(21,27)(23,30)(24,31)(28,29)(32,34);(3,6,7,5)(4,8,9,10)(12,15)(16,18)(20,25)(24,32)(26,33)(31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7417654);