Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 37748736 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C42 |
| direct product | C2×#2840 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C26 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,7)(8,10)(11,12)(13,16)(14,15)(17,18)(21,25)(23,26);(5,9)(8,10)(12,15)(16,17)(19,20,22,24)(21,26,25,23);(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,18)(15,17)(19,21,22,25)(20,23,24,26);(3,5)(4,8)(6,9)(7,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,23)(22,24)(25,26)52 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29659);