Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | Aut(G) = #72284492 (order 2048) · Inn(G) = #353 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #353 |
| commutator | C42 | C2×C8 |
| Frattini, cosocle | #423 | C22 |
| socle | C2 | #1325 |
| prime radical | C23 | #101 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #423 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,5,10,19,26,6,11,20,27,13,8,18,24)(3,12,14,25,15,7,17,21,16,22,30,31,28,23,29,32);(1,3,13,28,6,16,5,15)(2,7,10,22,11,23,8,12)(4,17,19,30,20,29,18,14)(9,25,24,32,27,31,26,21)Polycyclic presentation:
2^8|a^2=d,b^2=eg,c^2=gh,d^2=f,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=deg,d^c=dgh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94569, #94570, #94573, … |
| maximal normal subgroup of | #94569, #94570, #94573, … |
| maximal quotient of | #93725, #93726, #93740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 361);