Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,22,24)(19,25,29,32)(20,30,28,23)(21,31,27,26);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,27,20,22,29,21,28)(18,23,31,32,24,30,26,25);(2,6)(5,7)(11,15)(13,16)(18,24)(23,30)(25,32)(26,31);(2,6)(5,7)(17,20,27,29,22,28,21,19)(18,25,31,23,24,32,26,30)Polycyclic presentation:
2^8|a^2=gh,b^2=eh,d^2=ef,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28084, #6457401, #6457416, … |
| maximal normal subgroup of | C2×#28084, #6457401, #6457416, … |
| maximal quotient of | #361192, #362404, #367733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21072);