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) = C25×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×#547 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,11)(8,16,18,12)(9,13)(10,14)(15,20,17,19);(1,2)(3,4)(5,7,10,9)(6,11,14,13)(8,15,18,17)(12,19,16,20);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,20)(14,16);(2,4)(5,9,10,7)(6,13,14,11)(8,17,18,15)(12,20,16,19)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=h,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 | #138885, #138887, #138893, … |
| maximal normal subgroup of | #138885, #138887, #138893, … |
| maximal quotient of | #219720, #219721, #219727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15394);