Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
328 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(19,25)(20,28)(22,29)(24,30)(26,31)(27,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,26)(20,24)(21,23)(22,27)(25,31)(28,30)(29,32);(17,19,20,25,21,27,28,32)(18,22,24,29,23,26,30,31);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,23)(22,26)(24,30)(29,31)Polycyclic presentation:
2^8|b^2=eg,c^2=f,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,c^b=ch,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6474223, #6474224, #6474229, … |
| maximal normal subgroup of | #6474223, #6474224, #6474229, … |
| maximal quotient of | #392522, #393129, #393134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26262);