Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 65536 |
| 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 | #2428 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,6,8,7)(17,18)(19,20)(21,24)(22,23);(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(18,23)(19,21);(2,3)(6,7)(10,14)(11,16)(17,19,22,21)(18,20,23,24);(1,3,5,2)(4,6,8,7)(9,11,13,16)(10,15,14,12)(17,20)(18,19)(21,23)(22,24);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,11,14,16)(17,21,22,19)(18,24,23,20)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,d^2=fh,e^2=h,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 | #6580803, #6580806, #6580866, … |
| maximal normal subgroup of | #6580803, #6580806, #6580866, … |
| maximal quotient of | #519023, #519047, #519081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29450);