Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#121 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
497 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,11,15,10,24)(7,16,18,27,20,33,19,12)(8,13,25,30,23,31,22,29)(17,32,35,26,34,28,21,36);(1,2)(3,4)(5,7,10,19,11,20,9,18)(6,12,14,16,15,27,24,33)(8,21,22,34,23,35,25,17)(13,28,30,36,31,32,29,26);(2,4)(5,8)(6,13)(7,17)(9,22)(10,25)(11,23)(12,26)(14,29)(15,31)(16,32)(18,21)(19,35)(20,34)(24,30)(27,36)(28,33);(5,9,11,10)(6,14,15,24)(7,18,20,19)(8,22,23,25)(12,16,27,33)(13,29,31,30)(17,21,34,35)(26,32,36,28)Polycyclic presentation:
2^8|a^2=ef,b^2=e,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^c=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #359392, #359393, #377547, … |
| maximal normal subgroup of | #359392, #359393, #377547, … |
| maximal quotient of | #242363, #242364, #242367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16589);