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 | |
966 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)(21,22,27,25)(23,30,24,28)(26,33,34,29)(31,35,32,36);(6,10)(11,16)(12,17)(18,20)(21,23,27,24)(22,28,25,30)(26,31,34,32)(29,35,33,36);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,24,27,23)(22,28,25,30)(26,32,34,31)(29,35,33,36);(1,3,4,2)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,25,27,22)(23,28,24,30)(26,33,34,29)(31,35,32,36);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,26,27,34)(22,29,25,33)(23,31,24,32)(28,35,30,36)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=h,e^2=fg,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×#34505, #7626106, #7626113, … |
| maximal normal subgroup of | C2×#34505, #7626106, #7626113, … |
| maximal quotient of | #513837, #513856, #513866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27493);