Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | Aut(G) = #72340198 (order 4096) · Inn(G) = #2668 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
436 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 105 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)(3,11)(4,15)(5,19)(6,10)(7,13)(8,17)(9,22)(12,30)(14,27)(16,25)(18,28)(20,23)(21,26)(24,31)(29,32);(1,3)(2,7)(4,16)(5,20)(6,14)(8,25)(9,27)(10,23)(11,19)(12,17)(13,22)(15,24)(18,32)(21,31)(26,29)(28,30);(7,23)(11,27)(24,30)(25,29);(1,4,6,18)(2,8,10,26)(3,12,14,31)(5,21,22,17)(7,24,23,30)(9,28,19,15)(11,29,27,25)(13,32,20,16)Polycyclic presentation:
2^8|d^2=h,e^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151326, #151328, #151330, … |
| maximal normal subgroup of | #151326, #151328, #151330, … |
| maximal quotient of | #331928, #331930, #331934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23423);