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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
988 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,13)(15,16);(1,3)(2,4)(5,7)(6,8)(10,13)(14,15)(17,18,22,20)(19,24,21,23);(2,6)(3,7)(9,10)(11,15)(12,13)(14,16)(17,19,22,21)(18,23,20,24);(1,2)(3,4)(5,6)(7,8)(9,11,12,16)(10,14,13,15)(17,20,22,18)(19,23,21,24);(1,4)(2,7)(3,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,21,22,19)(18,24,20,23)Polycyclic presentation:
2^8|b^2=fh,c^2=fh,d^2=gh,e^2=fh,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 | #224132, #224138, #224147, … |
| maximal normal subgroup of | #224132, #224138, #224147, … |
| maximal quotient of | #519185, #519197, #519251, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29558);