Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #481 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
203 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 55 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)(7,15,16,33)(8,19,20,36)(9,24,22,13)(11,26,27,17)(12,29,30,23)(18,31,25,34)(21,28,35,32);(1,2)(3,4)(8,20)(12,30)(15,33)(17,26)(19,36)(21,35)(23,29)(31,34);(2,4)(5,7,9,25,10,16,22,18)(6,11,13,32,14,27,24,28)(8,21,23,26,20,35,29,17)(12,31,19,33,30,34,36,15)Polycyclic presentation:
2^8|a^2=h,c^2=dg,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,e^b=eh,d^c=dh,e^c=egh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13399, #499053, #499055, … |
| maximal normal subgroup of | C2×#13399, #499053, #499055, … |
| maximal quotient of | #144715, #144751, #145025, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6387);