Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,7)(5,8)(10,13)(11,15)(17,18)(19,22)(20,21)(23,24);(1,2,6,7)(3,8,4,5)(10,13)(14,16)(17,19,21,24)(18,22,20,23);(1,3,6,4)(2,8,7,5)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,22)(21,24);(1,4,6,3)(2,5,7,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,22,24,23);(1,5)(2,3)(4,7)(6,8)Polycyclic presentation:
2^8|b^2=fh,c^2=f,d^2=fh,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 | C2×#59917, #7665831, #7666079, … |
| maximal normal subgroup of | C2×#59917, #7665831, #7666079, … |
| maximal quotient of | #531879, #531913, #532137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52905);