Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #476 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1864 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
203 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2,6,13,8,14,7,26)(3,15,18,27,20,31,19,30)(4,21,5,24,23,12,22,11)(9,17,28,16,29,32,10,25);(1,3)(2,9,14,29)(4,17)(5,25)(6,19)(7,18)(8,20)(10,26,28,13)(11,27,24,30)(12,31,21,15)(16,22)(23,32);(1,3)(2,10)(4,17)(5,25)(6,19)(7,18)(8,20)(9,26)(11,15)(12,27)(13,29)(14,28)(16,22)(21,30)(23,32)(24,31)Polycyclic presentation:
2^8|a^2=f,b^2=fg,d^2=g,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,d^b=dg,e^b=eg,f^b=fh,g^b=gh,d^c=dg,e^c=eg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12951, #494327, #494999, … |
| maximal normal subgroup of | C2×#12951, #494327, #494999, … |
| maximal quotient of | #135905, #135929, #136226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5939);