Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#106 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,16,19,18)(11,21,20,22)(12,17)(13,24)(14,23);(2,5)(6,8)(9,10,13,18,14,20,24,22,15,19,12,16,23,11,17,21);(1,3,4,7)(2,6,5,8)(9,11,12,22,14,10,17,16,15,20,13,21,23,19,24,18)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=d,d^2=g,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,f^b=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #134988, #134992, #135008, … |
| maximal normal subgroup of | #134988, #134992, #135008, … |
| maximal quotient of | #135648, #135650, #135670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5874);