Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C24 · Out(G) = C211 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
891 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,18)(19,23)(20,22)(21,24);(10,15)(14,16)(17,19,20,24)(18,21,22,23);(9,12,13,11)(10,16,15,14)(18,22)(21,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|b^2=g,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #6385418, #6385420, #6385426, … |
| maximal normal subgroup of | #6385418, #6385420, #6385426, … |
| maximal quotient of | #514172, #514202, #514458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27854);