Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22)(25,26,28,32,29,31,35,38)(27,30,34,37,33,36,39,40);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(25,27)(26,30)(28,34)(29,33)(31,36)(32,37)(35,39)(38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)(26,31)(27,33)(28,35)(37,40)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=d,f^2=h,b^a=bd,c^a=ce,e^a=eg,c^b=ch,f^c=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #93707, #97243, #97244, … |
| maximal normal subgroup of | #93707, #97243, #97244, … |
| maximal quotient of | #99120, #99121, #99140, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1693);