Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2198 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24 |
| prime radical | C2×#453 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
622 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 622 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,13)(6,10)(7,11)(8,12)(9,14)(15,17)(16,18);(3,5,8,6)(4,9,12,17)(7,14,16,15)(10,11,13,18);(3,6,8,5)(4,10,12,13)(7,14,16,15)(9,11,17,18)40 conjugacy classes — view the full character table.
| maximal subgroup of | #152555, #153118, #153136, … |
| maximal normal subgroup of | #152555, #153118, #153136, … |
| maximal quotient of | C2×#12092, C2×#12097, C2×#12204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26533);