Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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 | |
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:
(2,5)(3,7)(9,10,14,12)(11,16,15,13)(17,18)(19,23)(20,22)(21,24);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,21)(18,23,20,24);(1,2)(3,6)(4,5)(7,8)(9,10,14,12)(11,13,15,16)(17,20,22,18)(19,23,21,24);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,16,15,13)(17,21,22,19)(18,23,20,24);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,15,12,11)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=fg,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #7035329, #7035330, #7035353, … |
| maximal normal subgroup of | #7035329, #7035330, #7035353, … |
| maximal quotient of | #531689, #531756, #531873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52314);