Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#1305 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #95 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4,5,8,6,7,9,10)(11,12)(13,17)(14,16)(15,18);(4,7)(5,9)(11,13)(12,15,16,17)(14,18)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1305, #105471, #105707, … |
| maximal normal subgroup of | C4×#1305, #105471, #105707, … |
| maximal quotient of | C2×#7032, #95110, #95124, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1091);