Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,24)(19,27,29,25)(20,23,30,32)(21,31,28,26);(2,5)(6,8)(17,19,28,30,22,29,21,20)(18,23,31,25,24,32,26,27);(9,11,12,16)(10,13,14,15)(18,24)(19,29)(20,30)(26,31);(1,3,4,7)(2,6,5,8)(17,20,21,29,22,30,28,19)(18,25,26,23,24,27,31,32)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=fh,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,c^b=ch,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #103010, #103020, #104638, … |
| maximal normal subgroup of | #103010, #103020, #104638, … |
| maximal quotient of | #392488, #393412, #393575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26287);