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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,21,23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(19,23)(21,24);(10,14)(13,15)(17,19,20,23)(18,21,22,24);(2,5)(6,8)(11,16)(13,15);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=fh,c^2=h,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 | #6378319, #6378320, #6378321, … |
| maximal normal subgroup of | #6378319, #6378320, #6378321, … |
| maximal quotient of | #520781, #520822, #520886, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45265);