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×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
868 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,8,7,5)(9,10,13,11)(12,14,16,15)(17,18,20,22)(19,23,24,21);(1,3)(2,5)(4,8)(6,7)(12,16)(14,15)(17,18,20,22)(19,21,24,23);(9,11,13,10)(12,15,16,14)(17,18)(19,21)(20,22)(23,24);(1,4,6,2)(3,8,7,5)(9,12)(10,14)(11,15)(13,16)(17,19,20,24)(18,21,22,23);(1,5,6,8)(2,7,4,3)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=h,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 | #7091685, #7091691, #7091701, … |
| maximal normal subgroup of | #7091685, #7091691, #7091701, … |
| maximal quotient of | #531802, #531945, #532064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52672);