Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | Aut(G) = #72339316 (order 4096) · Inn(G) = #535 · Out(G) = D42 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2487 |
| prime radical | #532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,8,14,7,12)(3,15,18,9,19,31,21,28)(4,20,22,10,17,32,26,29)(5,23,24,11,25,30,16,27);(1,3,6,18,8,19,7,21)(2,9,12,15,14,28,13,31)(4,16,22,5,17,24,26,25)(10,11,20,23,29,27,32,30);(1,4)(2,10)(3,16)(5,18)(6,22)(7,26)(8,17)(9,27)(11,28)(12,20)(13,32)(14,29)(15,30)(19,24)(21,25)(23,31);(1,5,6,24,8,25,7,16)(2,11,12,23,14,27,13,30)(3,17,18,26,19,4,21,22)(9,20,15,29,28,32,31,10)Polycyclic presentation:
2^8|a^2=eg,b^2=e,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #486701, #486702, #500168, … |
| maximal normal subgroup of | #486701, #486702, #500168, … |
| maximal quotient of | #314967, #314968, #314969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20682);