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) = C26×C4 |
| direct product | C2×#2809 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
959 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 959 conjugacy classes of subgroups — too many to draw. Only its 503 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 — 5 generators on 26 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,14,16)(13,17,18,15)(21,22)(24,25);(4,7)(5,10)(11,12,14,16)(13,15,18,17)(19,20,23,26)(21,24,22,25);(3,5)(4,8)(6,10)(7,9)(11,12)(13,15)(14,16)(17,18)(19,21,23,22)(20,24,26,25);(3,4)(5,8)(6,7)(9,10)(11,13,14,18)(12,15,16,17)(19,22,23,21)(20,25,26,24)64 conjugacy classes — view the full character table.
| maximal subgroup of | #97546, #131463, #149486, … |
| maximal normal subgroup of | #97546, #131463, #149486, … |
| maximal quotient of | C2×#14446, C2×#14464, C2×#14467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45257);