Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| 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 | |
966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(7,8)(9,10)(11,18)(12,16)(13,15)(14,19)(17,21)(20,23)(22,24);(1,2,5,6)(3,7,4,8)(9,11,12,19)(10,14,16,18)(13,20,21,24)(15,22,17,23);(1,3,5,4)(2,7,6,8)(10,15)(13,21)(14,22)(16,17)(18,23)(20,24);(1,4,5,3)(2,7,6,8)(11,19)(14,18)(20,24)(22,23);(10,16)(11,19)(15,17)(20,24)Polycyclic presentation:
2^8|b^2=f,c^2=fh,d^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=df,e^b=eh,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #7148104, #7148105, #7148115, … |
| maximal normal subgroup of | #7148104, #7148105, #7148115, … |
| maximal quotient of | #6379473, #6379476, #6380299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54412);