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 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17)(11,18,20,24)(12,16,14,22)(15,19,23,21);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,22)(13,17)(14,16)(18,21)(19,24)(20,23);(9,11,12,19,13,20,14,21)(10,15,22,24,17,23,16,18);(3,8)(5,7)(9,12,13,14)(10,16,17,22)(11,19,20,21)(15,18,23,24);(2,6)(5,7)(10,17)(11,20)(16,22)(19,21)Polycyclic presentation:
2^8|a^2=h,c^2=gh,d^2=h,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×#62318, #10577613, #10577658, … |
| maximal normal subgroup of | C2×#62318, #10577613, #10577658, … |
| maximal quotient of | #6386800, #6389161, #6389214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55306);