Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| 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 | C2×C4 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #336 |
| prime radical | C23 | #97 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
103 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)(9,10,13,11,12,16,20,18,15,17,14,19,22,23,21,24);(2,5)(3,7)(9,11,14,16,12,18,21,17,15,19,13,23,22,24,20,10)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=g,e^2=fg,g^2=h,b^a=bc,c^a=cf,e^a=eh,d^b=dfh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7210, #127156, #127157, … |
| maximal normal subgroup of | C2×#7210, #127156, #127157, … |
| maximal quotient of | #93279, #93344, #93352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 198);