Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(5,7)(10,14)(15,16)(17,18,22,25)(19,26,23,20)(21,24,29,31)(27,32,30,28);(1,2,6,4)(3,5,8,7)(19,23)(20,26)(21,29)(24,31);(1,3,6,8)(2,7,4,5)(9,10,13,14)(11,15,12,16)(17,19,22,23)(18,20,25,26)(21,27,29,30)(24,28,31,32);(1,4,6,2)(3,5,8,7)(9,11,13,12)(10,15,14,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,30)(27,31)(29,32);(1,5)(2,3)(4,8)(6,7)(9,12,13,11)(10,16,14,15)(17,21,22,29)(18,24,25,31)(19,27,23,30)(20,28,26,32)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=fg,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46219, #7702744, #7702852, … |
| maximal normal subgroup of | C2×#46219, #7702744, #7702852, … |
| maximal quotient of | #557183, #557193, #557479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39207);