Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | Aut(G) = C22×#72210469 (order 4096) · Inn(G) = #127 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #392 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
280 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,29,32,28);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,22,30)(18,23,26,32)(20,29,21,28)(24,31,25,27);(6,15)(9,16)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10016, #233569, #233571, … |
| maximal normal subgroup of | C2×#10016, #233569, #233571, … |
| maximal quotient of | #102544, #102546, #102554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3004);