Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(6,10)(7,13)(12,17)(14,19)(21,22)(23,26)(24,29)(25,30)(27,31)(28,33)(32,34)(35,36);(6,10)(11,16)(12,17)(18,20)(22,23)(24,31)(28,30)(32,36);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,26,22)(24,29,31,27)(25,30,33,28)(32,35,36,34);(1,3,4,2)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25,26,33)(22,28,23,30)(24,32,31,36)(27,34,29,35)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#34501, #7626106, #7626137, … |
| maximal normal subgroup of | C2×#34501, #7626106, #7626137, … |
| maximal quotient of | #513835, #513859, #513916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27489);