Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
330 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)(17,18,22,24,21,20,29,30)(19,25,28,32,27,23,31,26);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,28,21,27,29,31)(18,23,24,26,20,25,30,32);(3,7)(6,8)(10,14)(13,16)(17,20)(18,21)(19,26)(22,24)(23,28)(25,31)(27,32)(29,30);(1,3,4,7)(2,6,5,8)(18,20)(23,25)(24,30)(26,32)Polycyclic presentation:
2^8|a^2=g,b^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#30211, #6435849, #6435859, … |
| maximal normal subgroup of | C2×#30211, #6435849, #6435859, … |
| maximal quotient of | #380166, #380169, #380214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23199);