Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#2683 |
| 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 | |
1719 conjugacy classes of subgroups; only the 583 normal subgroups are shown.
This group has 1719 conjugacy classes of subgroups — too many to draw. Only its 583 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,6)(12,15)(16,17);(8,10)(11,12,14,15)(13,17,18,16);(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18);(3,4)(5,6)(11,13,14,18)(12,16,15,17)64 conjugacy classes — view the full character table.
| maximal subgroup of | #97536, #131458, #149477, … |
| maximal normal subgroup of | #97536, #131458, #149477, … |
| maximal quotient of | C2×#14362, C2×#14401, C2×#14440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45090);