Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | D4 |
| commutator | C4 | C82 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #336 |
| prime radical | C23 | #97 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(9,10,12,17,13,16,20,23,14,18,21,24,11,15,19,22);(1,2,3,5,4,6,7,8)(10,15,18,16)(17,22,24,23)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=f,e^2=g,f^2=h,b^a=bc,c^a=ch,e^a=eh112 conjugacy classes — view the full character table.
| maximal subgroup of | #92864, #92865, C2×#7138, … |
| maximal normal subgroup of | #92864, #92865, C2×#7138, … |
| maximal quotient of | #92897, #92899, #92904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 126);