Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(11,15)(13,16)(17,18,22,25)(19,21,27,23)(20,28,30,24)(26,31,32,29);(2,5)(3,7)(9,10,12,14)(11,13,15,16)(17,19,22,27)(18,23,25,21)(20,26,30,32)(24,31,28,29);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,29)(30,32);(1,3,4,7)(2,6,5,8)(17,20,22,30)(18,24,25,28)(19,26,27,32)(21,29,23,31);(1,2)(3,6)(4,5)(7,8)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,28,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#45981, #7702628, #7702705, … |
| maximal normal subgroup of | C2×#45981, #7702628, #7702705, … |
| maximal quotient of | #557042, #557055, #557387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38969);