Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10,14,12)(11,15,13,16)(17,18,21,19)(20,23,24,22);(1,2)(3,8)(4,5)(6,7)(10,12)(11,13)(17,19,21,18)(20,23,24,22);(2,5)(6,8)(9,11,14,13)(10,15,12,16)(17,20,21,24)(18,22,19,23);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,15,13,16);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,15,12,16)Polycyclic presentation:
2^8|a^2=fg,b^2=g,c^2=fg,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7119045, #7119053, #7119059, … |
| maximal normal subgroup of | #7119045, #7119053, #7119059, … |
| maximal quotient of | #535455, #535514, #536059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37972);