Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| 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 | #397 |
| 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:
(5,6,10,14,11,15,9,16)(7,17,21,29,20,28,19,12)(8,22,25,32,26,13,24,31)(18,30,34,27,33,35,23,36);(1,2)(3,4)(5,7,9,19,11,20,10,21)(6,12,14,17,15,29,16,28)(8,23,24,33,26,34,25,18)(13,30,31,27,22,35,32,36);(2,4)(5,8,11,26)(6,13,15,22)(7,18,20,33)(9,25,10,24)(12,27,29,36)(14,32,16,31)(17,30,28,35)(19,34,21,23)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=h,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,e^b=eh,d^c=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11134, #357823, C4×#2082, … |
| maximal normal subgroup of | C2×#11134, #357823, C4×#2082, … |
| maximal quotient of | #116990, #117018, #117020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4122);