Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#2607 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1199 conjugacy classes of subgroups; only the 519 normal subgroups are shown.
This group has 1199 conjugacy classes of subgroups — too many to draw. Only its 519 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) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(11,12)(13,17)(14,15)(16,18)(19,20,23,22)(21,25,26,24);(4,7)(8,10)(12,15)(16,17)(21,26)(24,25);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,21,23,26)(20,24,22,25);(3,5,6,9)(4,8,7,10)(11,13)(12,16)(14,18)(15,17)(19,22,23,20)(21,25,26,24)76 conjugacy classes — view the full character table.
| maximal subgroup of | #122164, #131453, #131454, … |
| maximal normal subgroup of | #122164, #131453, #131454, … |
| maximal quotient of | C2×#13953, C2×#14091, C2×#14093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27893);