Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#1971 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
787 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 787 conjugacy classes of subgroups — too many to draw. Only its 287 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,6)(7,8,9,10)(11,12)(13,21)(14,17)(15,24)(16,20)(18,22)(19,25)(23,26);(3,4,5,6)(8,10)(11,13,15,14,16,22,25,23)(12,17,19,21,20,26,24,18);(11,14,15,22,16,23,25,13)(12,18,19,17,20,21,24,26)64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7111, #124652, #147428, … |
| maximal normal subgroup of | C2×#7111, #124652, #147428, … |
| maximal quotient of | C2×#9658, C2×#9664, #204292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14213);