Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| 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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,8)(6,7)(10,14)(11,16)(18,21)(22,24);(1,2,5,8)(3,4,7,6)(9,10,13,14)(11,15,16,12)(17,18)(19,22)(20,21)(23,24);(1,3,5,7)(2,6,8,4)(9,11)(10,15)(12,14)(13,16);(1,2)(3,6)(4,7)(5,8);(1,4,5,6)(2,7,8,3)(9,12,13,15)(10,11,14,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|b^2=g,c^2=h,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#35100, #7628307, #7628348, … |
| maximal normal subgroup of | C2×#35100, #7628307, #7628348, … |
| maximal quotient of | #516406, #516420, #516622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28088);