Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
| direct product | C2×#2785 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1143 conjugacy classes of subgroups; only the 463 normal subgroups are shown.
This group has 1143 conjugacy classes of subgroups — too many to draw. Only its 463 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);(3,4,7,5)(6,9,10,8)(11,12)(13,16)(14,15)(17,18)(20,21)(24,25);(4,5)(6,10)(12,14)(16,17)(19,20,23,21)(22,25,26,24);(3,5,7,4)(6,9,10,8)(11,13,15,18)(12,16,14,17)(19,21,23,20)(22,25,26,24);(3,6,7,10)(4,8,5,9)(11,14,15,12)(13,17,18,16)(19,22)(20,24)(21,25)(23,26)64 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2785⋊C3 |
| maximal subgroup of | #131563, C2×#12145, C2×#12239, … |
| maximal normal subgroup of | #131563, C2×#12145, C2×#12239, … |
| maximal quotient of | C2×#14399, C2×#14476, C2×#14589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45215);