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 | C2×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,21,23,24);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,21)(20,22)(23,24);(2,6)(3,8)(9,10,12,14)(11,13,16,15);(1,3,5,8)(2,7,6,4)(9,10)(11,15)(12,14)(13,16)(17,19,20,23)(18,21,22,24);(1,4,5,7)(2,3,6,8)(9,11,12,16)(10,13,14,15)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=fgh,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 | #6581595, #6581604, #6581608, … |
| maximal normal subgroup of | #6581595, #6581604, #6581608, … |
| maximal quotient of | #519045, #519047, #519088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29515);