Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13691 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2206 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(3,13)(4,7)(5,17)(9,23)(10,16)(11,12)(14,25)(15,30)(18,21)(19,27)(20,26)(22,29)(24,32)(28,31);(1,2,7,11,8,12,4,6)(3,9,22,28,15,24,14,21)(5,10,19,26,20,27,16,17)(13,18,25,32,30,31,29,23);(1,3)(2,9)(4,14)(5,18)(6,21)(7,22)(8,15)(10,25)(11,28)(12,24)(13,17)(16,23)(19,32)(20,31)(26,30)(27,29)Polycyclic presentation:
2^8|b^2=dh,d^2=h,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ef,f^a=fg,g^a=gh,e^c=ef,f^c=fg,g^c=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | #151524, #151525, #151528, … |
| maximal normal subgroup of | #151524, #151525, #151528, … |
| maximal quotient of | #151620, #151621, #151628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6680);