Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33657 |
| automorphism | Aut(G) = #72314018 (order 4096) · Inn(G) = C2×#481 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#481 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3060 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
273 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 273 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)(4,9)(5,18)(6,16)(7,11)(8,15)(10,21)(13,27)(14,29)(17,19)(20,26)(22,32)(23,30)(24,31)(25,28);(1,2)(3,8)(4,16)(5,19)(6,9)(7,11)(10,22)(12,28)(13,26)(14,25)(15,29)(17,18)(20,30)(21,32)(23,24)(27,31);(1,3,6,23,7,14,4,13)(2,8,9,24,11,25,16,26)(5,20,21,28,22,31,17,15)(10,27,18,29,19,30,32,12);(1,4,7,6)(2,9,11,16)(3,13,14,23)(5,17,22,21)(8,24,25,26)(10,18,19,32)(12,27,29,30)(15,31,28,20)Polycyclic presentation:
2^8|c^2=eg,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^b=fh,e^c=eh,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #486689, #486690, #486694, … |
| maximal normal subgroup of | #486689, #486690, #486694, … |
| maximal quotient of | #493920, #494270, #494488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26647);