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) = 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 | |
811 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10,12,14)(11,13,15,16)(18,22)(19,24);(1,2,6,5)(3,8,7,4)(17,18,21,22)(19,23,24,20);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,22)(21,24);(1,4,6,8)(2,7,5,3);(1,5,6,2)(3,8,7,4)(9,11,12,15)(10,13,14,16)(17,20,21,23)(18,19,22,24)Polycyclic presentation:
2^8|a^2=fg,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 | #122071, #122128, #122134, … |
| maximal normal subgroup of | #122071, #122128, #122134, … |
| maximal quotient of | #516495, #516571, #516661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28190);