Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2200 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
664 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 664 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)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18);(3,5,7,12)(4,8,10,15)(6,13,14,18)(9,11,16,17);(5,12)(6,14)(8,15)(11,17)34 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#2200, #495751, #495776, … |
| maximal normal subgroup of | C4×#2200, #495751, #495776, … |
| maximal quotient of | C2×#12093, C2×#12094, C2×#12099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26536);