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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,4,6,8)(9,10,13,12)(11,15,16,14)(18,21)(22,24);(1,3)(2,4)(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,21)(23,24);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24);(1,3)(2,4)(5,6)(7,8)(9,12,13,10)(11,15,16,14)(17,19,20,23)(18,22,21,24);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)Polycyclic presentation:
2^8|a^2=h,b^2=fg,d^2=f,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×#36169, #7629815, #7629845, … |
| maximal normal subgroup of | C2×#36169, #7629815, #7629845, … |
| maximal quotient of | #519003, #519027, #519072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29157);