Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C4⋊C4 |
| prime radical | C23 | C4×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7,4,6,8,3)(10,15,17,21)(11,18)(16,23,24,22)(19,20);(2,6)(3,7)(9,10,13,15,14,17,12,21)(11,16,20,22,18,24,19,23);(1,3,8,6,4,7,5,2)(9,11,12,19,14,18,13,20)(10,16,21,23,17,24,15,22)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=cg,c^b=ch,d^b=dg,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #128721, #128723, #128724, … |
| maximal normal subgroup of | #128721, #128723, #128724, … |
| maximal quotient of | #96893, #96894, #96895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1283);