Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | Aut(G) = #72281625 (order 2048) · Inn(G) = #329 · Out(G) = #120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #329 |
| commutator | C42 | C42 |
| Frattini, cosocle | C2×#97 | C22 |
| socle | C2 | #1295 |
| prime radical | C23 | #101 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C2×C4 ▷ C2 ▷ C1 | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,7,11,6,21)(3,12,13,8,15,22,16,24)(5,18,17,25,19,26,20,10)(14,29,28,27,31,23,30,32);(1,3,5,14,6,16,20,30,7,15,19,31,4,13,17,28)(2,8,10,23,9,22,18,32,11,24,25,29,21,12,26,27)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=h,e^2=g,g^2=h,b^a=bc,c^a=cf,e^a=efg,g^a=gh,c^b=cg,f^b=fh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94678, #94679, C2×#7267, … |
| maximal normal subgroup of | #94678, #94679, C2×#7267, … |
| maximal quotient of | #93379, #93440, #93551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 255);