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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
863 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)(11,15)(12,13)(14,16)(17,18,22,23)(19,24,20,21);(1,2)(3,8)(4,5)(6,7)(10,13)(14,15)(17,19,22,20)(18,21,23,24);(2,5)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,19)(18,21,23,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,19)(20,23)(22,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=fg,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 | #7005694, #7005706, #7005763, … |
| maximal normal subgroup of | #7005694, #7005706, #7005763, … |
| maximal quotient of | #535451, #535622, #536051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37966);