Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | Aut(G) = #72257774 (order 1024) · Inn(G) = #327 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #327 |
| commutator | C2×C8 | C2×C8 |
| Frattini, cosocle | C2×#97 | C22 |
| socle | C2 | #1270 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
114 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 35 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 — 2 generators on 32 points:
(1,2)(3,11,13,27,15,28,14,23)(4,16,19,8)(5,10)(6,9)(7,12,21,17,22,29,26,30)(18,24)(20,31,32,25);(1,3)(2,7)(4,17)(5,13)(6,15)(8,23)(9,22)(10,26)(11,25)(12,20)(14,18)(16,27)(19,30)(21,24)(28,31)(29,32)Polycyclic presentation:
2^8|a^2=d,c^2=fh,d^2=g,f^2=h,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=def,f^b=fh,g^b=gh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7109, #124166, #124262, … |
| maximal normal subgroup of | C2×#7109, #124166, #124262, … |
| maximal quotient of | #92819, #92820, #92835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 97);