Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#2863 |
| 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 | |
1071 conjugacy classes of subgroups; only the 559 normal subgroups are shown.
This group has 1071 conjugacy classes of subgroups — too many to draw. Only its 559 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);(3,4,8,7)(5,10,6,9)(11,12,15,18)(13,16,17,14)(21,25)(23,26);(3,5,8,6)(4,9,7,10)(11,13,15,17)(12,16,18,14)(19,20,22,24)(21,23,25,26);(3,6,8,5)(4,10,7,9)(11,14,15,16)(12,17,18,13);(3,7,8,4)(5,10,6,9)(11,12)(13,16)(14,17)(15,18)(19,21)(20,23)(22,25)(24,26)88 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2863⋊C3 |
| maximal subgroup of | #96935, C2×#7669, C2×#7673, … |
| maximal normal subgroup of | #96935, C2×#7669, C2×#7673, … |
| maximal quotient of | C2×#14688, C2×#14699, C2×#14703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29621);