Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#2401 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator, Frattini, cosocle | C23 | C25 |
| socle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1147 conjugacy classes of subgroups; only the 519 normal subgroups are shown.
This group has 1147 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);(3,4)(5,8)(6,7)(9,10)(11,12,14,16)(13,15,17,18)(19,20)(21,23)(22,24)(25,26);(4,7)(8,10)(11,13,14,17)(12,15,16,18)(19,21,22,25)(20,23,24,26);(12,16)(15,18)(20,24)(21,25);(3,5,6,9)(4,8,7,10)88 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2401⋊C3 |
| maximal subgroup of | #372795, #372797, #372799, … |
| maximal normal subgroup of | #372795, #372797, #372799, … |
| maximal quotient of | C2×#13842, C2×#13846, C2×#13855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27537);