Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#121 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,22)(9,14)(10,15)(11,16)(12,20)(13,24)(18,32)(19,27)(21,30)(23,29)(25,28)(26,35)(31,34)(33,36);(1,3,4,2)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,26)(14,17)(15,30)(16,27)(22,31)(23,33)(24,34)(25,35)(28,36)(29,32);(5,8,10,24,11,25,9,23)(6,13,14,22,16,29,15,28)(7,18,20,34,21,35,19,33)(12,26,17,31,27,32,30,36);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,25,24)(12,17,27,30)(13,22,29,28)(18,33,35,34)(26,31,32,36)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #147187, #147190, #147203, … |
| maximal normal subgroup of | #147187, #147190, #147203, … |
| maximal quotient of | #203710, #203714, #203718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14169);