Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | #2535 | 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:
(1,2)(3,7)(4,5)(6,8)(9,10,13,16)(11,15,12,14)(18,21)(22,24);(1,3,6,4)(2,7,8,5)(9,11,13,12)(10,14,16,15);(2,8)(3,4)(9,10,13,16)(11,14,12,15)(17,18)(19,22)(20,21)(23,24);(1,4,6,3)(2,5,8,7)(9,10)(11,14)(12,15)(13,16)(17,19,20,23)(18,22,21,24);(1,5)(2,3)(4,8)(6,7)(9,12,13,11)(10,15,16,14)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6482378, #6482379, #6482380, … |
| maximal normal subgroup of | #6482378, #6482379, #6482380, … |
| maximal quotient of | #556426, #556428, #556446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30561);