Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(9,10,14,19)(11,17,12,21)(13,23,15,18)(16,22,24,20);(9,11,13,20,14,12,15,22)(10,16,18,17,19,24,23,21);(1,2,3,4)(9,12,14,11)(10,17,19,21)(13,22,15,20)(16,18,24,23)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=e,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11339, #358485, #358486, … |
| maximal normal subgroup of | C2×#11339, #358485, #358486, … |
| maximal quotient of | #121331, #121333, #121395, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4327);