Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,16,22)(11,23,27,38)(12,28,24,40)(13,31,34,19)(14,35,15,36)(17,26,39,25)(18,29,37,30)(20,33,21,32);(1,2,4,6)(3,8,7,5)(9,11,16,27)(10,17,22,39)(12,13,24,34)(14,29,15,30)(18,19,37,31)(20,40,21,28)(23,36,38,35)(25,33,26,32);(9,12)(10,18)(11,13)(14,26)(15,25)(16,24)(17,19)(20,38)(21,23)(22,37)(27,34)(28,36)(29,32)(30,33)(31,39)(35,40);(9,13,16,34)(10,19,22,31)(11,24,27,12)(14,32,15,33)(17,37,39,18)(20,35,21,36)(23,28,38,40)(25,30,26,29)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309715, #72309719, #72309730, … |
| maximal normal subgroup of | #72309715, #72309719, #72309730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396236);