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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,7,4,5)(10,13)(14,16)(17,18,21,24)(19,25,27,22)(20,28,30,23)(26,31,29,32);(3,4)(5,7)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31);(1,3,6,4)(2,5,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,22)(20,26)(21,27)(23,31)(24,25)(28,32)(29,30);(1,4,6,3)(2,7,8,5)(17,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,30)(29,32);(1,5,6,7)(2,3,8,4)(9,11,12,15)(10,14,13,16)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fg,d^2=fh,e^2=g,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×#46202, #7702777, #7702847, … |
| maximal normal subgroup of | C2×#46202, #7702777, #7702847, … |
| maximal quotient of | #557211, #557223, #557545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39190);