Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| 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 | |
839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,7)(3,5,8,4)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(1,3,6,8)(2,4,7,5)(17,19)(18,21)(20,23)(22,24);(1,4,6,5)(2,3,7,8)(10,13)(14,16)(18,22)(19,23);(2,7)(3,8)(10,13)(14,16)(18,22)(21,24);(1,5,6,4)(2,8,7,3)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df64 conjugacy classes — view the full character table.
| maximal subgroup of | #6408025, #6408026, #6408028, … |
| maximal normal subgroup of | #6408025, #6408026, #6408028, … |
| maximal quotient of | #520950, #520954, #520957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45286);