Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
621 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,14,9,15)(7,17,20,27,21,12,19,28)(8,13,23,31,24,30,25,32)(18,33,22,36,34,26,35,29);(5,7,11,21)(6,12,14,17)(8,22,24,35)(9,20,10,19)(13,29,30,36)(15,28,16,27)(18,25,34,23)(26,32,33,31);(2,4)(5,8)(6,13)(7,18)(9,25)(10,23)(11,24)(12,26)(14,30)(15,32)(16,31)(17,33)(19,35)(20,22)(21,34)(27,36)(28,29)Polycyclic presentation:
2^8|a^2=g,b^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,g^b=gh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11015, #357567, #358179, … |
| maximal normal subgroup of | C2×#11015, #357567, #358179, … |
| maximal quotient of | #111394, #111395, #111403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4003);