Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#2658 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4,5,6)(12,16)(13,19)(18,24)(21,22);(7,8,9,10)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,25)(23,24);(3,4)(5,6)(7,8)(9,10)(11,13)(12,17)(14,22)(15,19)(16,20)(18,26)(21,23)(24,25);(11,14,15,23)(12,18,16,24)(13,21,19,22)(17,25,20,26)64 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27618);