Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,18)(13,22)(14,17)(16,19)(20,23)(21,24);(1,3,5,8)(2,6,4,7)(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);(1,4,5,2)(3,6,8,7)(10,17)(15,23)(16,24)(18,22);(9,12,14,13)(10,18,17,22)(11,19,20,21)(15,16,23,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=dh,g^a=gh,d^b=df,e^b=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6574706, #6574707, #6574708, … |
| maximal normal subgroup of | #6574706, #6574707, #6574708, … |
| maximal quotient of | #6379396, #6379397, #6380134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54368);