Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10390 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| commutator, Frattini, prime radical, cosocle | C25 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
263 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 45 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:
(1,2,5,9)(3,7,14,23)(4,15)(6,11,19,10)(8,17)(12,27)(13,22,29,21)(16,24)(18,25)(20,30)(26,31)(28,32);(1,3)(2,7)(4,16,17,12)(5,14)(6,13)(8,24,15,27)(9,23)(10,21)(11,22)(18,30,31,28)(19,29)(20,25,32,26);(1,4,6,18)(2,8,11,26)(3,12,13,28)(5,17,19,31)(7,20,22,24)(9,15,10,25)(14,16,29,30)(21,27,23,32)Polycyclic presentation:
2^8|a^2=dg,b^2=d,c^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fgh,c^b=cf,e^b=eh,f^b=fg,d^c=dg28 conjugacy classes — view the full character table.
| maximal subgroup of | #151463, C2×#10393, #238075, … |
| maximal normal subgroup of | #151463, C2×#10393, #238075, … |
| maximal quotient of | #104759, #104760, #104763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3381);