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 | #392 | 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:
(2,5)(6,8)(9,10,14,12)(11,16,15,13)(17,18,20,21)(19,23,24,22);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(18,21)(22,23);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,16,15,13)(17,18)(19,22)(20,21)(23,24);(1,2)(3,6)(4,5)(7,8)(17,19)(18,22)(20,24)(21,23);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,15,12,11)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^8|a^2=f,c^2=fgh,e^2=fg,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×#59553, #7666486, #7666675, … |
| maximal normal subgroup of | C2×#59553, #7666486, #7666675, … |
| maximal quotient of | #532000, #532176, #532214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52541);