Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 6291456 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×Q8×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2707 conjugacy classes of subgroups; only the 1119 normal subgroups are shown.
This group has 2707 conjugacy classes of subgroups — too many to draw. Only its 1119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(11,12,14,15)(13,17,18,16);(11,13,14,18)(12,16,15,17);(4,6)(7,8,9,10)(11,15,14,12)(13,17,18,16);(3,4)(5,6)(7,8)(9,10)100 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×C22⋊C4×SL(2,𝔽3) |
| maximal subgroup of | #97578, #105988, #106661, … |
| maximal normal subgroup of | #97578, #105988, #106661, … |
| maximal quotient of | C2×C22⋊C4×C4⋊C4, C2×Q8×#95, C2×#13817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27088);