Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | Aut(G) = C22×#511819 (order 2048) · Inn(G) = C2×#124 · Out(G) = #120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2398 |
| prime radical | #575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 109 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)(3,12,15,25)(4,9)(5,10)(6,20)(7,11)(8,16,22,13)(14,26,29,31)(17,23)(18,24)(19,30)(21,28,32,27);(3,13,15,16)(4,17)(8,12,22,25)(9,23)(14,28,29,27)(18,19)(21,31,32,26)(24,30);(1,3)(2,8)(4,14)(5,13)(6,16)(7,15)(9,21)(10,25)(11,22)(12,20)(17,29)(18,27)(19,28)(23,32)(24,31)(26,30);(1,4,5,18,7,17,6,19)(2,9,10,24,11,23,20,30)(3,14,13,27,15,29,16,28)(8,21,25,31,22,32,12,26)Polycyclic presentation:
2^8|a^2=e,b^2=e,d^2=ef,f^2=h,g^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=dh,e^c=eh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #486084, #486085, C2×#23834, … |
| maximal normal subgroup of | #486084, #486085, C2×#23834, … |
| maximal quotient of | #244449, #244450, #244451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16822);