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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,8,5,7)(17,18)(19,22)(20,21)(23,24);(1,3,6,5)(2,7,4,8)(11,15)(13,16)(17,19,21,24)(18,22,20,23);(1,4,6,2)(3,8,5,7)(9,10,12,14)(11,13,15,16)(17,19)(18,23)(20,22)(21,24);(9,11,12,15)(10,13,14,16)(17,20,21,18)(19,22,24,23);(1,5,6,3)(2,8,4,7)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=fgh,d^2=h,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 | #7091176, #7091194, #7091239, … |
| maximal normal subgroup of | #7091176, #7091194, #7091239, … |
| maximal quotient of | #532115, #532122, #532245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52951);