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 | #3511 | 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)(19,26)(20,24)(21,29)(22,25)(23,28)(27,32)(30,31);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(19,27)(23,31)(26,32)(28,30);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19)(18,23)(20,28)(21,30)(22,27)(24,32)(25,31)(26,29);(2,8)(6,9)(7,16)(14,15)(17,18)(19,23)(20,29)(21,24)(22,25)(26,28)(27,31)(30,32);(17,20,22,21)(18,24,25,29)(19,28,27,30)(23,32,31,26)Polycyclic presentation:
2^8|b^2=f,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 | #249855, #249856, #249857, … |
| maximal normal subgroup of | #249855, #249856, #249857, … |
| maximal quotient of | #6374396, #6374470, #6374977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53874);