Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| 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 | |
849 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,13,16)(11,15,12,14)(17,18)(19,21)(20,22)(23,24);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,19)(18,21)(20,23)(22,24);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,15,16,14)(19,23)(21,24);(18,22)(21,24);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #96960, #122016, #122101, … |
| maximal normal subgroup of | #96960, #122016, #122101, … |
| maximal quotient of | #515075, #515108, #515171, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27583);