Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| 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 | #394 |
| 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 | |
619 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)(5,6,10,15,11,14,9,26)(7,16,19,31,20,12,18,28)(8,21,25,32,23,13,24,30)(17,27,35,29,34,33,22,36);(5,7,11,20)(6,12,14,16)(8,22,23,35)(9,19,10,18)(13,29,21,36)(15,31,26,28)(17,25,34,24)(27,32,33,30);(2,4)(5,8)(6,13)(7,17)(9,25)(10,24)(11,23)(12,27)(14,21)(15,32)(16,33)(18,35)(19,22)(20,34)(26,30)(28,36)(29,31)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,e^a=eh,f^a=fh,c^b=cf,f^b=fh,g^b=gh,d^c=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11035, #357591, C4×#2049, … |
| maximal normal subgroup of | C2×#11035, #357591, C4×#2049, … |
| maximal quotient of | #111296, #111297, #111407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4023);