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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
996 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)(3,8)(4,6)(5,7)(9,10,13,15)(11,12,14,16)(17,18,21,24)(19,22,20,23);(1,3)(2,6)(4,5)(7,8)(9,11)(10,12)(13,14)(15,16)(17,18)(19,22)(20,23)(21,24);(1,2,5,7)(3,8,4,6)(9,12)(10,14)(11,15)(13,16)(17,19,21,20)(18,22,24,23);(1,2)(3,6)(4,8)(5,7)(9,11)(10,12)(13,14)(15,16)(17,19,21,20)(18,22,24,23);(1,4,5,3)(2,6,7,8)(9,12)(10,14)(11,15)(13,16)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^8|a^2=fh,c^2=f,d^2=fg,e^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 | #223020, #223034, #223058, … |
| maximal normal subgroup of | #223020, #223034, #223058, … |
| maximal quotient of | #519202, #519222, #519255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29192);