Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 65536 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,16)(11,12,15,14)(17,18,20,22)(19,21,23,24);(1,2)(3,4)(5,6)(7,8)(9,11,13,15)(10,14,16,12)(17,18)(19,21)(20,22)(23,24);(2,6)(3,7)(9,10)(11,14)(12,15)(13,16);(1,3,5,7)(2,4,6,8);(1,4)(2,7)(3,6)(5,8)(9,12,13,14)(10,15,16,11)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6455164, #6455166, #6455178, … |
| maximal normal subgroup of | #6455164, #6455166, #6455178, … |
| maximal quotient of | #515318, #515351, #515430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28535);