Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | #2246 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(10,11)(12,13)(14,24)(15,30)(17,21)(18,33)(19,32)(20,23)(22,35)(25,39)(26,29)(27,28)(31,38)(34,37)(36,40);(2,5)(6,8)(9,10,13,17,14,18,27,32,30,35,39,40,38,37,26,23,16,20,29,34,31,36,25,22,15,19,28,33,24,21,12,11);(1,3,4,7)(2,6,5,8)(9,11,12,21,24,33,28,19,15,22,25,36,31,34,29,20,16,23,26,37,38,40,39,35,30,32,27,18,14,17,13,10)Polycyclic presentation:
2^8|b^2=e,c^2=d,d^2=fg,e^2=f,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg,g^a=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #151542, #151543, #151544, … |
| maximal normal subgroup of | #151542, #151543, #151544, … |
| maximal quotient of | #152384, #152830, #152888, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6707);