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 | #2537 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(4,7)(9,10,13,14)(11,16,12,15);(1,2)(3,7)(4,8)(5,6)(10,14)(15,16)(19,23)(21,24);(9,10)(11,15)(12,16)(13,14)(17,18,20,22)(19,21,23,24);(1,3,5,8)(2,7,6,4)(9,11,13,12)(10,15,14,16)(17,19)(18,21)(20,23)(22,24);(1,4,5,7)(2,8,6,3)(9,12,13,11)(10,16,14,15)Polycyclic presentation:
2^8|a^2=g,c^2=h,d^2=f,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 | #7011357, #7011369, #7011428, … |
| maximal normal subgroup of | #7011357, #7011369, #7011428, … |
| maximal quotient of | #531975, #532026, #532276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52386);