Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,14,9,8,10,13)(7,12,18,20,16,15,17,19)(22,26,28,31)(27,35,36,34);(5,7,9,16)(6,12,8,15)(10,18,11,17)(13,20,14,19)(21,22)(23,27)(24,31)(25,28)(26,33)(29,34)(30,36)(32,35);(2,4)(5,8,9,6)(7,15,16,12)(10,13,11,14)(17,19,18,20)(21,23,25,30)(22,27,28,36)(24,32,33,29)(26,34,31,35)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg,f^c=fg40 conjugacy classes — view the full character table.
| Fitting subgroup of | #8509⋊C3 |
| maximal subgroup of | #94545, C2×#8509, #187611, … |
| maximal normal subgroup of | #94545, C2×#8509, #187611, … |
| maximal quotient of | #97003, #97007, #97183, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1497);