Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(6,8)(9,10,14,18)(11,17,13,15)(12,16,21,24)(19,23,20,22);(9,11,14,13)(10,15,18,17)(12,20,21,19)(16,23,24,22);(5,6)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24);(9,12,14,21)(10,16,18,24)(11,19,13,20)(15,22,17,23);(1,3,4,2)(9,13,14,11)(10,17,18,15)(12,20,21,19)(16,23,24,22)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| Fitting subgroup of | #34291⋊C3 |
| maximal subgroup of | #6383912, #6383917, #6383923, … |
| maximal normal subgroup of | #6383912, #6383917, #6383923, … |
| maximal quotient of | #513623, #513651, #513692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27279);