Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | C2×D4 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
115 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)(3,5)(4,7)(6,8)(9,10,14,18)(11,19,21,33)(12,24,26,39)(13,27,25,17)(15,28,29,22)(16,31,32,20)(23,38,37,35)(30,36,34,40);(9,11,20,36,13,23,26,28,14,21,31,40,25,37,12,22)(10,15,24,35,17,30,32,33,18,29,39,38,27,34,16,19);(2,5)(3,8)(6,7)(11,21)(15,29)(19,33)(22,28)(23,37)(30,34)(35,38)(36,40)Polycyclic presentation:
2^8|a^2=h,b^2=df,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,c^b=ch,e^c=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | #151522, #151523, #151552, … |
| maximal normal subgroup of | #151522, #151523, #151552, … |
| maximal quotient of | #123385, #123386, #123391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5422);