Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,6,10,16,24)(3,11,14,25,12,21,27,31)(5,7,15,18,17,19,29,30)(8,20,22,26,23,28,32,13);(2,7)(3,12)(4,15)(8,21)(10,19)(11,23)(13,26)(14,28)(16,29)(20,27)(22,32)(25,31);(1,3)(2,8,7,11)(4,14)(5,13)(6,12)(9,22,18,25)(10,23,19,21)(15,20)(16,27)(17,26)(24,32,30,31)(28,29)Polycyclic presentation:
2^8|a^2=f,c^2=d,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,f^b=fg,e^c=eg46 conjugacy classes — view the full character table.
| maximal subgroup of | #132139, #132140, #132141, … |
| maximal normal subgroup of | #132139, #132140, #132141, … |
| maximal quotient of | #99154, #99155, #99158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1724);