Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#1270 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
497 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 135 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 — 3 generators on 26 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12)(13,15)(14,17)(16,18)(19,20,21,23,22,24,25,26);(3,5)(4,7)(6,10)(8,9)(12,15)(13,18)(16,17)76 conjugacy classes — view the full character table.
| maximal subgroup of | #94086, #94307, C4×#1270, … |
| maximal normal subgroup of | #94086, #94307, C4×#1270, … |
| maximal quotient of | C2×#7025, C4×#1270, #96910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 546);