Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C24 · Out(G) = C211 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,6,4,7)(9,10,13,15)(11,16,12,14)(17,18,20,22)(19,21,23,24);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,16)(13,15)(19,23)(21,24);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,20,23)(18,21,22,24);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,14,15,16);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^8|a^2=fh,c^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df64 conjugacy classes — view the full character table.
| maximal subgroup of | #6378755, #6378757, #6378759, … |
| maximal normal subgroup of | #6378755, #6378757, #6378759, … |
| maximal quotient of | #520786, #520825, #520883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45273);