Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#2240 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 143 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 — 4 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(12,18)(13,23)(14,25)(15,26)(16,19)(20,21)(22,24);(4,7)(8,10)(11,12,15,20,16,21,14,18,17,22,25,13,19,23,26,24);(3,5,6,9)(4,8,7,10)(11,13,14,12,19,18,15,23,17,20,26,22,16,24,25,21)76 conjugacy classes — view the full character table.
| maximal subgroup of | #137505, #137507, #137565, … |
| maximal normal subgroup of | #137505, #137507, #137565, … |
| maximal quotient of | C2×#12888, C2×#13147, C2×#13202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26682);