Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#397 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,24)(4,8)(5,9)(6,10)(7,13,21,18)(12,27,15,26)(16,22)(17,23)(19,25)(20,29,31,28)(30,32);(1,3)(2,7)(4,15)(5,18)(6,14)(8,20)(9,24)(10,21)(11,25)(12,16)(13,19)(17,28)(22,31)(23,27)(26,32)(29,30);(1,4,5,17,6,16,19,30)(2,8,9,23,10,22,25,32)(3,12,13,28,14,15,18,29)(7,20,11,26,21,31,24,27);(4,16)(8,22)(12,15)(17,30)(20,31)(23,32)(26,27)(28,29)Polycyclic presentation:
2^8|a^2=f,c^2=e,e^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #395789, #395790, #395819, … |
| maximal normal subgroup of | #395789, #395790, #395819, … |
| maximal quotient of | #242916, #242917, #242918, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16637);