Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| 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 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
973 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)(3,7)(9,10)(11,21)(12,23)(13,18)(14,20)(15,19)(16,17)(22,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,23)(14,17)(15,22)(18,24)(20,21);(9,12,15,16,14,24,13,21)(10,17,18,23,20,11,19,22);(2,5)(6,8)(9,13,14,15)(10,18,20,19)(11,22,17,23)(12,21,24,16);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,19,20,18)(11,22,17,23)(12,16,24,21)Polycyclic presentation:
2^8|c^2=g,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,e^c=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62461, #10579221, #10579237, … |
| maximal normal subgroup of | C2×#62461, #10579221, #10579237, … |
| maximal quotient of | #6386902, #6389533, #6389552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55449);