Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
261 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18,20,24,22,25,21,30)(19,26,28,32,27,23,29,31);(1,3)(2,7,9,16)(4,13)(5,11)(6,12)(8,10,14,15)(17,19,21,29,22,27,20,28)(18,23,24,31,25,26,30,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=g,e^2=h,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #105776, #105871, #106896, … |
| maximal normal subgroup of | #105776, #105871, #106896, … |
| maximal quotient of | #95904, #95905, #95906, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1074);