Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
385 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,22,27,31)(20,23,28,24)(26,29,32,30);(2,5)(6,8)(9,11,12,15)(10,13,14,16);(10,14)(13,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(1,3,4,7)(2,6,5,8)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,g^d=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6470529, #6470530, #6470531, … |
| maximal normal subgroup of | #6470529, #6470530, #6470531, … |
| maximal quotient of | #392364, #392737, #392740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26089);