Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,13,20,26)(7,11,17,22,16,23,29,33)(8,12,19,25,18,24,30,34)(15,21,28,32,27,31,35,36);(2,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,21)(13,23)(14,22)(18,27)(19,28)(20,29)(24,31)(25,32)(26,33)(30,35)(34,36);(5,8,9,19,10,18,20,30)(6,12,14,25,13,24,26,34)(7,15,17,28,16,27,29,35)(11,21,22,32,23,31,33,36);(8,18)(12,24)(15,27)(19,30)(21,31)(25,34)(28,35)(32,36);(6,13)(7,16)(12,24)(14,26)(15,27)(17,29)(25,34)(28,35)Polycyclic presentation:
2^8|a^2=g,c^2=g,g^2=h,b^a=bf,e^a=eh,e^b=eh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #129357, #129363, #129364, … |
| maximal normal subgroup of | #129357, #129363, #129364, … |
| maximal quotient of | #6345605, #6345606, #6345607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53131);