Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8528 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,10,5,4,6,9)(3,8,14,16,12,11,13,15)(18,23)(19,26)(20,21)(25,28)(27,29)(30,31);(1,3,5,12)(2,8,4,11)(6,14,7,13)(9,16,10,15)(17,18,22,25)(19,24,27,32)(20,28,21,23)(26,31,29,30);(1,4,5,2)(3,11,12,8)(6,9,7,10)(13,15,14,16)(17,19)(18,24)(20,26)(21,29)(22,27)(23,30)(25,32)(28,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg,f^c=fgh40 conjugacy classes — view the full character table.
| Fitting subgroup of | #8533⋊C3 |
| maximal subgroup of | #94549, C2×#8533, #187898, … |
| maximal normal subgroup of | #94549, C2×#8533, #187898, … |
| maximal quotient of | #97047, #97051, #97268, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1521);