Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(17,18,20,21)(19,23,24,22);(9,10,12,14)(11,15,16,13)(18,21)(22,23);(1,3,5,8)(2,6,4,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,24);(17,19)(18,22)(20,24)(21,23);(1,4,5,2)(3,7,8,6)(9,11,12,16)(10,13,14,15)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#39291, #7636961, #7637078, … |
| maximal normal subgroup of | C2×#39291, #7636961, #7637078, … |
| maximal quotient of | #521608, #522240, #522866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32279);