Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#576 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,15,11,7,10,14)(8,13,19,12,20,17,18,16);(5,7)(6,11)(8,16)(9,14)(10,15)(12,20)(13,19)(17,18);(2,4)(5,8)(6,12)(7,16)(9,18)(10,19)(11,20)(13,15)(14,17);(5,8)(6,13)(7,17)(9,19)(10,18)(11,20)(12,15)(14,16)Polycyclic presentation:
2^8|a^2=g,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=de,e^b=eh,f^c=fh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #497430, #498124, #501862, … |
| maximal normal subgroup of | #497430, #498124, #501862, … |
| maximal quotient of | #220220, #220226, #220234, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15451);