Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 196608 |
| 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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(10,15)(11,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28);(9,10,14,15)(11,12,16,13)(19,27)(21,31)(23,32)(25,28);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30);(9,12,14,13)(10,16,15,11)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,11,15,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)Polycyclic presentation:
2^8|a^2=h,b^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.
| Fitting subgroup of | #35504⋊C3 |
| maximal subgroup of | #133790, #254477, #254499, … |
| maximal normal subgroup of | #133790, #254477, #254499, … |
| maximal quotient of | #515499, #515660, #516173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28492);