Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(9,10)(11,14)(12,16)(13,15)(17,18,20,22)(19,23,24,21);(1,2,5,4)(3,6,8,7)(10,11)(15,16)(17,18,20,22)(19,23,24,21);(9,11,14,10)(12,16,13,15)(17,19,20,24)(18,21,22,23);(1,3,5,8)(2,6,4,7)(9,12,14,13)(10,15,11,16)(17,18)(19,21)(20,22)(23,24);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,16,11,15)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #98609, #251545, #251546, … |
| maximal normal subgroup of | #98609, #251545, #251546, … |
| maximal quotient of | #531662, #531674, #531962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52237);