Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) = #72282918 (order 2048) · Inn(G) = #442 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #442 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#442 |
| prime radical | #3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
326 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 149 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 — 4 generators on 32 points:
(1,2)(3,14,17,24)(4,18)(5,11)(6,10)(7,22)(8,13)(9,15,26,32)(12,20)(16,29,23,30)(19,28)(21,25,31,27);(1,3)(2,9)(4,15)(5,16)(6,21)(7,23)(8,17)(10,24)(11,25)(12,29)(13,26)(14,28)(18,30)(19,31)(20,32)(22,27);(1,4,7,6,8,20,5,19)(2,10,11,18,13,28,22,12)(3,15,23,21,17,32,16,31)(9,24,25,30,26,14,27,29);(1,5,8,7)(2,11,13,22)(3,16,17,23)(4,19,20,6)(9,25,26,27)(10,18,28,12)(14,29,24,30)(15,31,32,21)Polycyclic presentation:
2^8|a^2=f,c^2=fg,d^2=h,e^2=g,g^2=h,b^a=be,c^a=cg,e^a=eg,g^a=gh,e^b=egh,f^b=fh,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #493247, #493248, #494999, … |
| maximal normal subgroup of | #493247, #493248, #494999, … |
| maximal quotient of | #483589, #483590, #483591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26456);