Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
| direct product | C2×#2972 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C2 | C24×C8 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C23×C4 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,13,15,14,16,17,18);(3,5)(4,7)(6,10)(8,9);(5,10)(7,9);(4,8)(7,9)136 conjugacy classes — view the full character table.
| maximal subgroup of | #105610, #105612, #105648, … |
| maximal normal subgroup of | #105610, #105612, #105648, … |
| maximal quotient of | C2×#20523, C2×#20533, C2×#20555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53208);