Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7339 |
| automorphism | Aut(G) = #72237745 (order 2048) · Inn(G) = #1316 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1316 |
| commutator | C22×C4 | C2×C8 |
| Frattini, cosocle | #381 | C22 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
178 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 23 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 — 2 generators on 32 points:
(1,2,5,9)(3,11,13,29,14,24,30,32)(4,15,17,27)(6,10,19,28)(7,20,21,16,22,31,26,12)(8,23,25,18);(1,3)(2,7)(4,16)(5,13)(6,14)(8,24)(9,26)(10,22)(11,15)(12,18)(17,31)(19,30)(20,23)(21,28)(25,29)(27,32)Polycyclic presentation:
2^8|a^2=d,c^2=h,d^2=f,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ch,d^b=deh,f^b=fh,d^c=dg,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94522, #94523, #94622, … |
| maximal normal subgroup of | #94522, #94523, #94622, … |
| maximal quotient of | #93192, #93203, #93673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 327);