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 | |
883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(4,8)(6,7)(10,14)(12,15)(17,18,22,21)(19,24,20,23);(2,3)(6,7)(10,14)(11,16)(17,19,22,20)(18,23,21,24);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,20,22,19)(18,24,21,23);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,15,14,12);(1,4)(2,6)(3,7)(5,8)(9,12,13,15)(10,16,14,11)(17,21,22,18)(19,24,20,23)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=gh,d^2=h,e^2=g,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 | #122072, #206973, #206974, … |
| maximal normal subgroup of | #122072, #206973, #206974, … |
| maximal quotient of | #516498, #516574, #516660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28227);