Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | #2617 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(18,21)(19,20);(2,6)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,21)(19,24,20,23);(9,10,12,14)(11,13,16,15)(17,19,22,20)(18,23,21,24);(1,3,5,7)(2,4,6,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,19)(18,24,21,23);(1,4)(2,7)(3,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,21,22,18)(19,24,20,23)Polycyclic presentation:
2^8|b^2=g,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#56519, #8099119, #8099274, … |
| maximal normal subgroup of | C2×#56519, #8099119, #8099274, … |
| maximal quotient of | #585868, #586983, #587175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49507);