Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| 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 | #3008 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1049 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,15)(13,17)(14,16)(18,22)(20,24)(21,23);(2,5)(6,8)(9,11,13,12)(10,15,17,19)(14,20,22,21)(16,23,18,24);(10,16)(14,22)(15,23)(17,18)(19,24)(20,21);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,17,19)(14,21,22,20)(16,23,18,24);(11,12)(15,19)(20,21)(23,24)Polycyclic presentation:
2^8|b^2=h,d^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,e^b=eh,g^c=gh,e^d=eh61 conjugacy classes — view the full character table.
| maximal subgroup of | #6576087, #6576092, #6576099, … |
| maximal normal subgroup of | #6576087, #6576092, #6576099, … |
| maximal quotient of | #6370991, #6371819, #6371820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53580);