Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9431 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
245 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 245 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:
(1,2,5,12,17,28,3,15)(4,20,23,27,32,11,18,29)(6,13,16,8,7,14,19,10)(9,26,25,22,30,31,24,21);(2,8,13,12)(3,16)(4,21)(5,19)(9,24,11,27)(10,14,15,28)(20,29,30,25)(22,32);(2,9)(3,16)(4,22)(5,19)(8,24)(10,25)(11,13)(12,27)(14,20)(15,29)(18,23)(21,32)(26,31)(28,30)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #153307, #153308, C2×#9439, … |
| maximal normal subgroup of | #153307, #153308, C2×#9439, … |
| maximal quotient of | #100111, #100117, #100879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2427);