Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #72216677 (order 2048) · Inn(G) = C42 · Out(G) = #2010 |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C42 |
| commutator | C4 | C4×C16 |
| Frattini, cosocle | C82 | C22 |
| socle, prime radical | C22 | #322 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
61 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,7,4,9,11,15,5,8,10,16,12,6,13,14);(2,6,8,9)(3,10)(7,15,16,14)(11,13)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32)Polycyclic presentation:
2^8|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=h,b^a=be,d^a=dg,c^b=cg88 conjugacy classes — view the full character table.
| maximal subgroup of | #93668, #93759, C2×#7053, … |
| maximal normal subgroup of | #93668, #93759, C2×#7053, … |
| maximal quotient of | #92896, #93667, #93668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41);