Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#107 |
| prime radical | C23 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,21,16,23,29,22)(11,18,25,36,26,37,27,32)(12,17,24,33,28,35,40,34)(13,20,14,19,30,38,31,39);(2,5)(6,8)(9,11,14,24,15,25,30,28,16,26,31,40,29,27,13,12)(10,17,20,32,22,34,39,37,23,35,38,36,21,33,19,18);(1,3,4,7)(2,6,5,8)(9,12,13,27,29,40,31,26,16,28,30,25,15,24,14,11)(10,18,19,33,21,36,38,35,23,37,39,34,22,32,20,17)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=d,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #94204, #94205, #94208, … |
| maximal normal subgroup of | #94204, #94205, #94208, … |
| maximal quotient of | #135651, #135652, #135675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5879);