Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #386 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 119 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 — 3 generators on 36 points:
(2,3)(5,6,10,14,11,16,9,15)(7,12,19,27,20,28,18,32)(8,21,23,13,24,30,25,31)(17,29,33,26,34,36,22,35);(1,2,4,3)(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,29)(14,27)(15,32)(16,28)(17,23)(21,35)(24,33)(25,34)(26,30)(31,36);(1,3,4,2)(5,8)(6,13)(7,17)(9,25)(10,23)(11,24)(12,26)(14,30)(15,21)(16,31)(18,22)(19,33)(20,34)(27,36)(28,35)(29,32)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cf,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11218, #358084, #358190, … |
| maximal normal subgroup of | C2×#11218, #358084, #358190, … |
| maximal quotient of | #120005, #120034, #120036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4206);