Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C26×C8 (order 512) · Inn(G) = C23 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C16 | C23 |
| commutator, socle | C22 | C22×C16 |
| prime radical | C2×D4 | C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,8,13,9,14,20,28,10,7,16,17,22,18,31,32,11,15,21,29,23,30,34,36,24,19,12,25,26,33,27,35);(2,4);(5,7,8,17,9,18,20,32,10,15,16,29,22,30,31,36,11,19,21,25,23,33,34,35,24,6,12,13,26,14,27,28)Polycyclic presentation:
2^8|a^2=e,c^2=e,e^2=f,f^2=g,g^2=h,b^a=bd,c^a=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13645, #512742, #512743, … |
| maximal normal subgroup of | C2×#13645, #512742, #512743, … |
| maximal quotient of | #151074, #151088, #151089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6633);