Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
422 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 127 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 — 4 generators on 36 points:
(5,6,12,16,11,18,10,17)(7,19,28,15,9,27,21,13)(8,22,24,14,26,32,25,31)(20,30,34,33,23,29,35,36);(1,2)(3,4)(5,7,10,21,11,9,12,28)(6,13,16,19,18,15,17,27)(8,20,24,34,26,23,25,35)(14,29,22,33,31,30,32,36);(2,4)(5,8)(6,14)(7,20)(9,23)(10,24)(11,26)(12,25)(13,29)(15,30)(16,22)(17,32)(18,31)(19,33)(21,34)(27,36)(28,35);(5,9,12,21,11,7,10,28)(6,15,17,19,18,13,16,27)(8,23,25,34,26,20,24,35)(14,30,32,33,31,29,22,36)Polycyclic presentation:
2^8|a^2=fh,b^2=e,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #509299, #509828, #510222, … |
| maximal normal subgroup of | #509299, #509828, #510222, … |
| maximal quotient of | #277933, #277936, #277939, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17795);