Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #95 |
| commutator, prime radical | C23 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #336 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,13,16,15,18,23,11,12,17,22,21,24,19,14,20);(2,5)(3,7)(9,11,14,19,24,16,22,17,12,20,23,18,15,21,13,10)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=g,e^2=fg,g^2=h,b^a=bc,c^a=cf,e^a=eh,c^b=ch,d^b=df52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7211, #127160, #127161, … |
| maximal normal subgroup of | C2×#7211, #127160, #127161, … |
| maximal quotient of | #93280, #93345, #93353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 199);