Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 131072 |
| 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 | |
897 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)(11,16)(12,15)(13,14)(17,18,20,22)(19,21,23,24);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24);(2,6)(3,8)(10,14)(11,15);(1,3,5,8)(2,7,6,4)(9,10)(11,12)(13,14)(15,16);(1,4,5,7)(2,3,6,8)(9,12)(10,15)(11,14)(13,16)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^8|a^2=fg,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 | #130574, #130892, #164366, … |
| maximal normal subgroup of | #130574, #130892, #164366, … |
| maximal quotient of | #519186, #519252, #519281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29497);