Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) = #72303434 (order 4096) · Inn(G) = C2×#99 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#99 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C2 | C2×#324 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
182 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,9,20,25,7,11,21,27,4,17,18,26)(3,12,15,29,16,23,30,32,13,28,19,31,14,8,22,24);(1,3)(2,8,11,23)(4,14)(5,19)(6,16)(7,13)(9,12,17,28)(10,24,27,32)(15,21)(18,30)(20,22)(25,29,26,31);(3,13)(8,23)(12,28)(14,16)(15,19)(22,30)(24,32)(29,31)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=h,e^2=g,g^2=h,b^a=bd,d^a=df,f^a=fh,c^b=ch,e^b=efh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #129265, #129274, #129305, … |
| maximal normal subgroup of | #129265, #129274, #129305, … |
| maximal quotient of | #122297, #122298, #122299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4444);