Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | #2819 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,4)(6,7)(9,10,13,14)(11,12,16,15)(17,18,21,19)(20,23,24,22);(1,2,5,8)(3,7,4,6)(10,14)(12,15)(17,19,21,18)(20,23,24,22);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,15,16,12)(17,20,21,24)(18,22,19,23);(1,4,5,3)(2,7,8,6)(9,11)(10,15)(12,14)(13,16);(1,2)(3,6)(4,7)(5,8)(9,12,13,15)(10,11,14,16)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fg,d^2=fh,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 | #98477, #260772, #260776, … |
| maximal normal subgroup of | #98477, #260772, #260776, … |
| maximal quotient of | #557083, #557113, #557361, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39150);