Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3512 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,25)(21,30)(22,26)(24,27)(28,32)(29,31);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,21)(18,23,26,30)(20,28,29,27)(24,25,32,31);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32);(2,8)(6,9)(7,16)(14,15)(17,20)(18,25)(19,28)(21,27)(22,29)(23,32)(24,30)(26,31);(17,19,22,21)(18,23,26,30)(20,27,29,28)(24,31,32,25)Polycyclic presentation:
2^8|b^2=fh,c^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cfh,g^c=gh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #249869, #249870, #249871, … |
| maximal normal subgroup of | #249869, #249870, #249871, … |
| maximal quotient of | #6374397, #6374471, #6374978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53875);