Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(9,10,14,13)(11,16,12,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30);(1,2,5,4)(3,6,7,8)(9,11,14,12)(10,15,13,16)(18,22)(19,25)(24,30)(26,27);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,16,13,15)(17,19,21,25)(18,23,22,29)(20,27,28,26)(24,32,30,31);(9,10,14,13)(11,15,12,16)(17,20,21,28)(18,24,22,30)(19,26,25,27)(23,31,29,32);(1,4,5,2)(3,8,7,6)(9,13,14,10)(11,16,12,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #132118, #253332, #253354, … |
| maximal normal subgroup of | #132118, #253332, #253354, … |
| maximal quotient of | #515504, #515508, #515657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28499);