Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,25,21,24,26,19)(20,28,31,27,30,23,32,29);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,26,24,21,25,22,18)(20,29,32,23,30,27,31,28);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,20,21,30)(18,23,24,28)(19,27,25,29)(22,31,26,32);(2,6)(5,7)(18,24)(19,25)(20,30)(31,32)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=f,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=ceh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #136157, #136174, #136233, … |
| maximal normal subgroup of | #136157, #136174, #136233, … |
| maximal quotient of | #355225, #355366, #355865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15286);