Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1895 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,24)(13,28)(14,31)(15,18)(16,23)(17,33)(20,21)(22,36)(25,34)(26,30)(27,29)(32,37)(35,38)(39,40);(2,5)(6,8)(9,11,15,23)(10,16,18,19)(12,25,27,35)(13,21,30,40)(14,32,33,22)(17,37,31,36)(20,28,39,26)(24,38,29,34);(1,3,4,7)(2,6,5,8)(9,12,15,27)(10,13,18,30)(11,20,23,39)(14,26,33,28)(16,34,19,38)(17,29,31,24)(21,37,40,36)(22,25,32,35);(9,13,15,30)(10,12,18,27)(11,21,23,40)(14,29,33,24)(16,35,19,25)(17,26,31,28)(20,37,39,36)(22,38,32,34);(2,5)(6,8)(9,14,15,33)(10,17,18,31)(11,22,23,32)(12,26,27,28)(13,29,30,24)(16,36,19,37)(20,25,39,35)(21,38,40,34)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422118);