Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9263 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #386 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
212 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(3,9)(4,12)(8,14)(10,11)(13,15)(17,18,21,23,22,26,20,25)(19,27,29,24,30,31,28,32);(1,2,4,6,5,8,12,14)(3,7,10,13,11,16,9,15)(18,23)(20,21)(24,27)(25,26)(28,29)(31,32);(1,3)(2,7)(4,10)(5,11)(6,13)(8,16)(9,12)(14,15)(17,19,22,30)(18,24,26,32)(20,29,21,28)(23,27,25,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=egh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9268, #198596, #198600, … |
| maximal normal subgroup of | C2×#9268, #198596, #198600, … |
| maximal quotient of | #99916, #99924, #100588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2256);