Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | Aut(G) = #72284990 (order 2048) · Inn(G) = #354 · Out(G) = #120 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #354 |
| commutator | C42 | C2×C8 |
| Frattini, cosocle | #423 | C22 |
| socle | C2 | #1324 |
| 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 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,10,6,11,20,28,7,12,19,29,5,9,21,26)(3,13,14,27,16,8,22,23,17,24,30,32,15,25,18,31);(1,3,5,15,7,17,6,16)(2,8,11,24,12,25,9,13)(4,18,20,14,19,22,21,30)(10,27,26,31,29,32,28,23)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=g,d^2=f,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=degh,d^c=dgh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94561, #94562, #94564, … |
| maximal normal subgroup of | #94561, #94562, #94564, … |
| maximal quotient of | #93719, #93737, #93946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 356);