Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 413 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:
(5,6,10,16,11,14,9,15)(7,17,20,27,21,12,19,28)(8,22,23,30,24,31,25,13)(18,32,33,26,34,36,35,29);(1,2)(3,4)(5,7,11,21)(6,12,14,17)(8,18,24,34)(9,20,10,19)(13,29,30,26)(15,28,16,27)(22,36,31,32)(23,35,25,33);(2,4)(5,8)(6,13,14,30)(7,18)(9,23)(10,25)(11,24)(12,26,17,29)(15,22,16,31)(19,33)(20,35)(21,34)(27,36,28,32)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #133506, #133507, #133508, … |
| maximal normal subgroup of | #133506, #133507, #133508, … |
| maximal quotient of | #118669, #118713, #118717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4144);