Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3,5,6,8,7)(9,10,15,20)(11,21,12,17)(13,19,14,18)(16,23,24,22);(9,11,15,12)(10,16,20,24)(13,22,14,23)(17,18,21,19);(1,3,4,6,5,7,8,2)(9,12,15,11)(10,17,20,21)(13,22,14,23)(16,18,24,19);(9,13,15,14)(10,18,20,19)(11,22,12,23)(16,21,24,17)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eg,d^c=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #222289, #222291, #222292, … |
| maximal normal subgroup of | #222289, #222291, #222292, … |
| maximal quotient of | #182435, #182450, #183621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10870);