Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#2424 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 14 points:
(1,2);(4,6)(11,12,13,14);(3,4)(5,6)(8,10)(11,12,13,14);(7,8)(9,10)(11,12)(13,14);(3,4)(5,6)76 conjugacy classes — view the full character table.
| maximal subgroup of | #221400, #221464, #372680, … |
| maximal normal subgroup of | #221400, #221464, #372680, … |
| maximal quotient of | C2×#13873, C2×#13899, C2×#13967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27638);