Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#2571 |
| 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 | |
983 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 983 conjugacy classes of subgroups — too many to draw. Only its 567 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);(3,4,7,5)(6,9,10,8)(12,14)(16,17)(19,20,23,26)(21,25,22,24);(4,5)(8,9)(11,12,15,14)(13,16,18,17)(19,21,23,22)(20,24,26,25);(3,5,7,4)(6,9,10,8)(11,13,15,18)(12,16,14,17)(19,20)(21,24)(22,25)(23,26);(3,6,7,10)(4,8,5,9)(11,14,15,12)(13,17,18,16)(19,22,23,21)(20,25,26,24)76 conjugacy classes — view the full character table.
| maximal subgroup of | #98516, #122063, #122094, … |
| maximal normal subgroup of | #98516, #122063, #122094, … |
| maximal quotient of | C2×#13911, C2×#13982, C2×#14030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27930);