Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,12)(13,21)(14,15)(16,19)(17,22)(18,23)(20,24);(5,7,8,6)(10,15)(12,16)(17,21)(20,23);(1,2)(3,4)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24);(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24);(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #105766, #105879, #106090, … |
| maximal normal subgroup of | #105766, #105879, #106090, … |
| maximal quotient of | #513549, #513571, #513576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27191);