Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61923 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6194 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);(3,4)(5,13)(6,10)(7,11)(8,12)(9,15)(14,18)(16,17)(19,20,25,21)(22,30,32,34)(23,27,24,29)(26,33,28,31);(3,5,8,15)(4,9,12,13)(6,14,7,16)(10,17,11,18)(19,21,25,20)(22,28,32,26)(23,27,24,29)(30,31,34,33);(3,6,8,7)(4,10,12,11)(5,14,15,16)(9,17,13,18)(19,22)(20,26)(21,28)(23,31)(24,33)(25,32)(27,34)(29,30);(3,7,8,6)(4,11,12,10)(5,14,15,16)(9,17,13,18);(4,12)(9,13)(10,11)(17,18)(19,23,25,24)(20,27,21,29)(22,31,32,33)(26,34,28,30)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485794);