Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10,13,11)(12,15,16,14)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(19,26)(23,30)(27,31)(28,32);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32);(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32);(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #36053⋊C3 |
| maximal subgroup of | #6579569, #6579572, #6579619, … |
| maximal normal subgroup of | #6579569, #6579572, #6579619, … |
| maximal quotient of | #517954, #517955, #517958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29041);