Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,25,20,26,18,24)(15,27,17,23,29,41,34,21)(16,22,32,39,33,40,31,38)(28,36,30,37,42,44,43,35);(1,2)(3,4)(13,15,20,29)(14,21,26,23)(16,30,33,43)(17,19,34,18)(22,36,40,44)(24,27,25,41)(28,31,42,32)(35,38,37,39);(2,4)(6,9)(10,12)(13,16,19,32,20,33,18,31)(14,22,25,39,26,40,24,38)(15,28,17,30,29,42,34,43)(21,35,27,36,23,37,41,44);(2,4)(13,17,19,29,20,34,18,15)(14,23,25,41,26,21,24,27)(16,30,32,42,33,43,31,28)(22,37,39,44,40,35,38,36);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,24,26,25)(15,17,29,34)(16,31,33,32)(21,27,23,41)(22,38,40,39)(28,30,42,43)(35,36,37,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363854);