Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | Aut(G) = #72282918 (order 2048) · Inn(G) = #481 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #481 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1919 |
| prime radical | C23 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
126 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,7,12,5,11,8,22)(3,13,16,30,15,9,6,20)(4,17,18,24,19,32,23,29)(10,25,26,31,28,21,27,14);(1,3,7,16,5,15,8,6)(2,9,12,20,11,13,22,30)(4,14,18,25,19,31,23,21)(10,24,26,32,28,29,27,17);(1,4,7,18,5,19,8,23)(2,10,22,27,11,28,12,26)(3,14,16,25,15,31,6,21)(9,24,30,17,13,29,20,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=dg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^b=eh,d^c=dh,e^c=egh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13115, #494809, #494999, … |
| maximal normal subgroup of | C2×#13115, #494809, #494999, … |
| maximal quotient of | #140075, #140093, #146900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6103);