Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C2×#553 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(1,2,5,6)(3,7,4,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=e,d^2=e,b^a=be,c^a=cf,f^a=fh,d^b=de,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #139305, #139307, #139312, … |
| maximal normal subgroup of | #139305, #139307, #139312, … |
| maximal quotient of | #224490, #224492, #224494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15523);