Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7333 |
| automorphism | Aut(G) = #244122 (order 512) · Inn(G) = #322 · Out(G) = D4 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #322 |
| commutator | C8 | C4×C8 |
| Frattini, cosocle | #346 | C22 |
| socle | C2 | #1313 |
| prime radical | C22 | #347 |
| LCS | G ▷ C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #346 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,8,4,10,14,24,5,9,15,26,17,21,31,32,6,11,13,25,16,28,30,22,18,27,12,23,29,7,19,20);(2,7,9,10,11,21,27,28)(3,12,13,15)(4,16)(8,22,23,32,25,24,26,20)(14,31,30,19)(17,29)Polycyclic presentation:
2^8|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,g^2=h,b^a=be,d^a=dgh,f^a=fh,c^b=cgh,e^b=eh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7333, #149504, #149515, … |
| maximal normal subgroup of | C2×#7333, #149504, #149515, … |
| maximal quotient of | #93668, #93997, C2×#7333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 321);