Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 131072 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(10,13)(14,16)(17,18,22,21)(19,24,20,23);(2,5)(6,8)(17,19,22,20)(18,23,21,24);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,20,22,19)(18,24,21,23);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,21,22,18)(19,24,20,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=g,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #122129, #122156, #207023, … |
| maximal normal subgroup of | #122129, #122156, #207023, … |
| maximal quotient of | #516548, #516549, #516703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28292);