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, 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:
(2,6)(3,8)(9,10,14,12)(11,13,16,15);(2,6)(3,8)(9,11,14,16)(10,15,12,13)(19,23)(21,24);(1,2,5,6)(3,7,8,4)(9,12,14,10)(11,13,16,15)(17,18,20,22)(19,21,23,24);(1,3)(2,7)(4,6)(5,8)(17,19,20,23)(18,21,22,24);(1,4,5,7)(2,3,6,8)(9,13)(10,11)(12,16)(14,15)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=fh,d^2=gh,e^2=g,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×#59997, #7666583, #7666640, … |
| maximal normal subgroup of | C2×#59997, #7666583, #7666640, … |
| maximal quotient of | #532172, #532243, #532310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52985);