Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) = C4×#6628842 (order 2048) · Inn(G) = C2×D4 · Out(G) = C4×#120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C8×D4 |
| prime radical | C2×D8 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6)(3,5,8,7)(9,10,13,17,14,11,18,20,12,15,21,23,22,19,16,24);(2,5)(3,8)(6,7)(10,15)(11,19)(17,23)(20,24);(2,6)(5,7)(9,11,13,20,14,15,18,23,12,19,21,24,22,10,16,17)Polycyclic presentation:
2^8|a^2=e,c^2=ef,d^2=f,e^2=g,g^2=h,b^a=bd,c^a=cfh,d^a=df,c^b=ch,d^b=df64 conjugacy classes — view the full character table.
| maximal subgroup of | #122964, #122966, #122968, … |
| maximal normal subgroup of | #122964, #122966, #122968, … |
| maximal quotient of | #122964, #122965, #122966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5052);