Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| 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×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
911 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)(3,4)(5,6)(7,8)(11,15)(13,16)(18,21)(19,23);(2,6)(3,7)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,21)(23,24);(1,2,5,6)(3,4,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,20,23)(18,22,21,24);(1,3,5,7)(2,4,6,8);(1,4)(2,7)(3,6)(5,8)Polycyclic presentation:
2^8|b^2=fg,c^2=f,d^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #96977, #405722, #405723, … |
| maximal normal subgroup of | #96977, #405722, #405723, … |
| maximal quotient of | #516393, #516403, #516586, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28148);