Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
378 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)(3,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,31)(27,30)(28,32);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,25)(21,27)(23,30)(24,29)(26,28)(31,32);(1,4,5,6)(2,3,7,8)(10,14)(13,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(2,7)(3,8)(10,14)(13,16)(18,23)(22,30)(24,32)(29,31)Polycyclic presentation:
2^8|b^2=f,c^2=e,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6632329, #6632342, #6632345, … |
| maximal normal subgroup of | #6632329, #6632342, #6632345, … |
| maximal quotient of | #383715, #383944, #384488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24322);