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) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 419 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:
(1,2)(3,4)(5,6,10,14,11,16,9,15)(7,17,21,29,20,28,19,12)(8,22,25,13,26,30,24,31)(18,33,35,32,23,27,34,36);(5,7,9,19,11,20,10,21)(6,12,14,17,16,29,15,28)(8,23,24,35,26,18,25,34)(13,27,30,36,31,33,22,32);(2,4)(5,8,11,26)(6,13,16,31)(7,18,20,23)(9,25,10,24)(12,27,29,33)(14,22,15,30)(17,32,28,36)(19,35,21,34)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11143, #357839, C4×#2049, … |
| maximal normal subgroup of | C2×#11143, #357839, C4×#2049, … |
| maximal quotient of | #116996, #117038, #117042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4131);