Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
917 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24);(1,3,4,8)(2,5,6,7)(17,18)(19,23)(20,21)(22,24);(10,14)(15,16)(18,21)(22,23);(9,11,13,12)(10,15,14,16)(17,19,20,24)(18,22,21,23);(9,12,13,11)(10,16,14,15)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=g,e^2=gh,b^a=bf,c^a=cg,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #122091, #131484, #206759, … |
| maximal normal subgroup of | #122091, #131484, #206759, … |
| maximal quotient of | #514228, #514504, #514505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27966);