Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×C4⋊#95 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C23 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C24 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1927 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,6)(8,9)(15,16,18,17);(7,8,10,9)(11,12,13,14)(15,17,18,16);(3,4,5,6)(7,9,10,8)112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7039, #95955, #95966, … |
| maximal normal subgroup of | C2×#7039, #95955, #95966, … |
| maximal quotient of | C2×#7554, C4×C4⋊#95, C2×#7557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6742);