Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | Aut(G) = #72225052 (order 2048) · Inn(G) = #367 · Out(G) = C4×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #367 |
| commutator | C16 | C2×C8 |
| Frattini, cosocle | C4×C16 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,5,4,6,7,8)(10,15,17,28)(11,20)(12,24)(13,25)(16,31)(18,32,35,40)(19,38,36,30)(21,26)(22,34)(23,33)(27,39,37,29);(9,10,16,32,12,18,11,15,25,36,20,29,21,27,31,38,14,17,34,40,26,35,23,28,13,19,33,39,24,37,22,30)Polycyclic presentation:
2^8|a^2=d,b^2=ceg,c^2=egh,d^2=f,e^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94613, #94754, C2×#7463, … |
| maximal normal subgroup of | #94613, #94754, C2×#7463, … |
| maximal quotient of | #93884, #94023, #94754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 451);