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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(3,7)(9,10,13,16)(11,15,12,14);(1,2,5,4)(3,8,7,6)(9,11,13,12)(10,14,16,15)(19,23)(21,24);(1,3,5,7)(2,6,4,8)(17,18,20,22)(19,21,23,24);(1,2,5,4)(3,6,7,8)(17,19)(18,21)(20,23)(22,24);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,15,16,14)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=gh,d^2=g,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 | #7088105, #7088119, #7088170, … |
| maximal normal subgroup of | #7088105, #7088119, #7088170, … |
| maximal quotient of | #531919, #531944, #532219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52487);