Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8794 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #382 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,6,3)(4,5,8,7)(9,10)(11,19,22,15)(12,23,26,38)(13,18)(14,17)(16,25,31,35)(20,34)(21,33,37,29)(24,39)(27,32)(28,36)(30,40);(1,3,5,8,6,2,7,4)(9,11,13,21,14,22,27,37)(10,15,18,29,17,19,32,33)(12,24,25,40,26,20,35,36)(16,30,23,34,31,28,38,39);(1,4,5,3,6,8,7,2)(9,12)(10,16,17,31)(11,20)(13,25)(14,26)(15,28,19,30)(18,23,32,38)(21,36)(22,24)(27,35)(29,39,33,34)(37,40)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=d,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,f^b=fgh,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8794, #191228, #191229, … |
| maximal normal subgroup of | C2×#8794, #191228, #191229, … |
| maximal quotient of | #99282, #99289, #108607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1782);