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 | |
397 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 111 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)(7,16)(8,21,25,36)(9,26)(10,14)(11,15)(12,18)(13,24,29,23)(17,32,22,27)(19,28)(20,30)(31,34,35,33);(5,7,11,20)(6,12,15,28)(8,22,25,17)(9,19,10,18)(13,27,29,32)(14,30,26,16)(21,35,36,31)(23,33,24,34);(2,4)(5,8,9,23,11,25,10,24)(6,13,14,21,15,29,26,36)(7,17,18,33,20,22,19,34)(12,27,16,31,28,32,30,35)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11158, #357917, C4×#2082, … |
| maximal normal subgroup of | C2×#11158, #357917, C4×#2082, … |
| maximal quotient of | #118671, #118711, #118717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4146);