Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#2647 |
| 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 | |
927 conjugacy classes of subgroups; only the 527 normal subgroups are shown.
This group has 927 conjugacy classes of subgroups — too many to draw. Only its 527 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);(4,5)(8,9)(11,12)(13,16)(14,17)(15,18)(19,20,23,21)(22,25,26,24);(3,4,7,5)(6,8,10,9)(11,13)(12,14)(15,17)(16,18)(19,21,23,20)(22,24,26,25);(3,5,7,4)(6,8,10,9)(11,13)(12,16)(14,18)(15,17)(19,22,23,26)(20,24,21,25);(3,6,7,10)(4,8,5,9)(11,14,15,16)(12,17,18,13)64 conjugacy classes — view the full character table.
| maximal subgroup of | #97032, #97036, #122148, … |
| maximal normal subgroup of | #97032, #97036, #122148, … |
| maximal quotient of | C2×#14266, C2×#14276, C2×#14277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27604);