Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7035 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #95 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C22×C42 | C22 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,13,15,16)(12,14)(17,18,20,25,22,26,21,24)(19,27,28,23,30,31,29,32);(2,5)(3,8)(6,7)(9,11)(10,13,14,16)(12,15)(17,19,21,29,22,30,20,28)(18,23,25,31,26,32,24,27)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=fg,e^2=fg,b^a=bc,c^a=cf,e^a=efgh,c^b=cg,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93771, #93795, C2×#7044, … |
| maximal normal subgroup of | #93771, #93795, C2×#7044, … |
| maximal quotient of | #92809, #92812, #93622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32);