Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#95 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C25 | C23 |
| prime radical | C26 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1207 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 1207 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,9,11,14)(4,7)(5,8)(6,10,13,16)(12,15);(1,3)(2,6,8,14)(4,11)(5,10)(7,13,15,9)(12,16);(2,7)(6,13)(8,15)(9,14)Polycyclic presentation:
2^8|a^2=e,b^2=f,b^a=bd,c^a=cg,d^a=dg,f^a=fh,d^b=dh,e^b=eg52 conjugacy classes — view the full character table.
| Fitting subgroup of | #8101⋊C3 |
| maximal subgroup of | #106654, #106655, #106766, … |
| maximal normal subgroup of | #106654, #106655, #106766, … |
| maximal quotient of | #95112, #95115, #95124, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1089);