Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | Aut(G) = C22×#511819 (order 2048) · Inn(G) = #535 · Out(G) = C25 |
| 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 | #2598 |
| prime radical | #2017 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 101 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)(3,13)(4,17)(5,10)(6,11)(7,12)(8,15)(9,18)(14,22)(16,23)(19,26)(20,24)(21,27)(25,31)(28,30)(29,32);(1,3,5,15,7,16,6,20)(2,8,11,13,12,24,10,23)(4,14,18,29,19,31,21,30)(9,25,17,32,27,22,26,28);(1,4)(2,9)(3,14)(5,18)(6,21)(7,19)(8,22)(10,26)(11,17)(12,27)(13,28)(15,29)(16,31)(20,30)(23,32)(24,25);(2,10,12,11)(8,23,24,13)(9,26,27,17)(22,32,25,28)Polycyclic presentation:
2^8|b^2=e,d^2=g,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,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139698, #139699, #150743, … |
| maximal normal subgroup of | #139698, #139699, #150743, … |
| maximal quotient of | #326603, #326604, #326605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22374);