Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#577 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
675 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,14,11,7,9,15)(8,17,18,12,20,13,19,16);(5,7,11,6)(8,16,20,12)(9,14,10,15)(13,18,17,19);(2,4)(5,8)(6,12)(7,16)(9,18)(10,19)(11,20)(13,15)(14,17);(5,8)(6,13)(7,17)(9,19)(10,18)(11,20)(12,15)(14,16)Polycyclic presentation:
2^8|a^2=g,b^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,d^b=de,e^b=eh,f^c=fh,g^c=gh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22467, #6379205, #6379207, … |
| maximal normal subgroup of | C2×#22467, #6379205, #6379207, … |
| maximal quotient of | #220224, #220230, #220236, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15455);