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 | |
975 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)(9,10,15,19)(11,16,17,24)(12,21,23,22)(13,18,14,20);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,17)(18,23)(19,24);(9,12,14,16,15,23,13,24)(10,17,18,21,19,11,20,22);(2,5)(6,8)(9,13,15,14)(10,18,19,20)(11,21,17,22)(12,16,23,24);(1,3,4,7)(2,6,5,8)(11,17)(12,23)(16,24)(21,22)Polycyclic presentation:
2^8|a^2=h,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,d^c=dh,e^c=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62468, #10579228, #10579237, … |
| maximal normal subgroup of | C2×#62468, #10579228, #10579237, … |
| maximal quotient of | #6386905, #6389538, #6389555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55456);