Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12506 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
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)(19,27)(20,30)(21,29)(22,31)(23,28)(24,26)(25,32);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,27,18,23,29,22,25,24,20,30,26,32,31,21,28);(2,7)(6,14)(8,12)(9,15)(17,20,30,18,23,31,21,25,24,19,27,26,32,29,22,28)Polycyclic presentation:
2^8|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=gh,c^b=ch46 conjugacy classes — view the full character table.
| maximal subgroup of | #123323, #123324, #123325, … |
| maximal normal subgroup of | #123323, #123324, #123325, … |
| maximal quotient of | #135772, #141649, #141650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5500);