Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C22×C8×#120 (order 1024) · Inn(G) = C23 · Out(G) = C24×C8 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C16 | C23 |
| commutator, socle, prime radical | C22 | C22×C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,12,17,13,18,24,32,14,11,20,21,26,22,35,36,15,19,25,33,27,34,38,40,28,23,16,29,30,37,31,39);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(9,11,12,21,13,22,24,36,14,19,20,33,26,34,35,40,15,23,25,29,27,37,38,39,28,10,16,17,30,18,31,32)Polycyclic presentation:
2^8|a^2=e,b^2=d,c^2=de,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 | #94617, #151088, #151090, … |
| maximal normal subgroup of | #94617, #151088, #151090, … |
| maximal quotient of | #151984, #151985, #152790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6648);