Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(4,7)(10,13)(14,16)(17,18,21,24)(19,20,23,22);(2,6)(3,8)(17,19,21,23)(18,22,24,20);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24);(1,3,5,8)(2,7,6,4);(1,4,5,7)(2,8,6,3)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,23,24,19)Polycyclic presentation:
2^8|a^2=f,b^2=f,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 | #122070, #122127, #122133, … |
| maximal normal subgroup of | #122070, #122127, #122133, … |
| maximal quotient of | #516495, #516571, #516660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28180);