Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
420 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 420 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:
(1,2,4,3)(5,6,10,15,11,16,9,14)(7,17,20,12,21,28,19,29)(8,22,25,31,26,13,24,32)(18,33,34,36,35,30,23,27);(1,3,4,2)(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,30)(14,28)(15,17)(16,29)(18,24)(22,33)(25,35)(26,34)(27,31)(32,36);(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,27)(14,17)(15,28)(16,29)(22,36)(23,24)(25,34)(26,35)(30,31)(32,33);(5,8,9,24,11,26,10,25)(6,13,14,31,16,22,15,32)(7,18,19,23,21,35,20,34)(12,27,17,30,29,36,28,33)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,e^b=eh,g^b=gh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #498127, #498380, #504518, … |
| maximal normal subgroup of | #498127, #498380, #504518, … |
| maximal quotient of | #229185, #229191, #229198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16035);