Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9495 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C24 | C22×C4 |
| Frattini, prime radical, cosocle | C25 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
330 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 61 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)(3,7,12,20)(4,14,16,8)(5,9)(6,10)(11,19,25,31)(13,21,24,30)(15,18)(17,22,29,23)(26,28,27,32);(1,3,6,13)(2,7,18,30)(4,11,16,26)(5,12,15,24)(8,19,22,31)(9,20,10,21)(14,28,23,32)(17,27,29,25);(1,4,15,29)(2,8,18,22)(3,11,24,25)(5,17,6,16)(7,19,30,31)(9,23,10,14)(12,27,13,26)(20,32,21,28)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132862, #132864, #132866, … |
| maximal normal subgroup of | #132862, #132864, #132866, … |
| maximal quotient of | #100198, #100202, #100214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2490);