Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,21,23,20);(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,16,15)(17,19,22,23)(18,20,24,21);(2,6)(3,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,20)(21,23)(22,24);(1,3,5,8)(2,7,6,4)(9,11,12,16)(10,13,14,15)(17,20,22,21)(18,19,24,23);(1,4,5,7)(2,3,6,8)(9,10)(11,15)(12,14)(13,16)(17,21,22,20)(18,23,24,19)Polycyclic presentation:
2^8|a^2=g,b^2=f,d^2=gh,e^2=fh,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#36510, #7630201, #7630223, … |
| maximal normal subgroup of | C2×#36510, #7630201, #7630223, … |
| maximal quotient of | #519186, #519187, #519212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29498);