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 | #2407 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
967 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)(3,7)(4,5)(6,8)(10,14)(12,15)(18,21)(19,23);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(18,21)(19,23);(9,11,13,16)(10,15,14,12)(17,18,20,21)(19,22,23,24);(1,2)(3,6)(4,5)(7,8)(17,19)(18,22)(20,23)(21,24);(1,3,4,8)(2,6,5,7)(9,12,13,15)(10,16,14,11)Polycyclic presentation:
2^8|c^2=fh,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 | #7010848, #7010852, #7010864, … |
| maximal normal subgroup of | #7010848, #7010852, #7010864, … |
| maximal quotient of | #531977, #531978, #532298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52365);