Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| 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 | C2×#99 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
488 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 75 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)(7,15)(8,12)(9,22,23,13)(10,14)(11,17,25,32)(16,29)(18,30)(19,33)(20,35,36,27)(21,28)(24,31,26,34);(5,7,10,18)(6,11,14,25)(8,19,21,16)(9,17,23,32)(12,26,28,24)(13,15,22,30)(20,34,36,31)(27,33,35,29);(2,4)(5,8,10,21)(6,12,14,28)(7,16,18,19)(9,20,23,36)(11,24,25,26)(13,27,22,35)(15,29,30,33)(17,31,32,34)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch,f^b=fg,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #133415, #133417, #133419, … |
| maximal normal subgroup of | #133415, #133417, #133419, … |
| maximal quotient of | #129214, #129257, #129327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5680);