Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,7)(9,10)(11,15)(12,17)(13,22)(14,18)(16,19)(20,24)(21,23);(1,2,4,5)(3,7,8,6)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24);(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23);(1,3,4,8)(2,6,5,7);(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,24,23)Polycyclic presentation:
2^8|b^2=f,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=df,e^b=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #220661, #220663, #220689, … |
| maximal normal subgroup of | #220661, #220663, #220689, … |
| maximal quotient of | #6379392, #6379462, #6380333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54317);