Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,25,19)(20,29,30,22)(21,26,32,31)(23,28,24,27);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,29,26,23,28,21,30,25,18,22,31,24,27,32,20);(2,7)(6,14)(8,12)(9,15)(17,20,32,27,24,31,22,18,25,30,21,28,23,26,29,19)Polycyclic presentation:
2^8|a^2=h,b^2=dg,c^2=ef,d^2=fh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #123291, #123292, #123293, … |
| maximal normal subgroup of | #123291, #123292, #123293, … |
| maximal quotient of | #135770, #141648, #141666, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5498);