Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C8 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #340 |
| prime radical | C23 | #103 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,4,7,5,8,3,6)(10,14)(11,18)(12,19)(15,17)(16,23)(20,24)(21,22);(2,6,8,7)(9,10,11,15,19,21,24,22,13,17,20,14,12,16,18,23)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fgh,f^2=h,b^a=bc,c^a=cf,e^a=ef,f^a=fh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7176, #126781, #126842, … |
| maximal normal subgroup of | C2×#7176, #126781, #126842, … |
| maximal quotient of | #92940, #92944, #92956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 164);