Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,23,19,24,29,34)(15,25,27,21,28,37,16,20)(17,22,33,40,32,39,41,43)(26,38,31,36,42,44,30,35);(2,4)(13,15,19,28)(14,20,24,21)(16,29,27,18)(17,31,32,30)(22,38,39,44)(23,37,34,25)(26,41,42,33)(35,43,36,40);(6,9)(10,12)(13,16,18,15,19,27,29,28)(14,21,23,37,24,20,34,25)(17,31,33,42,32,30,41,26)(22,35,40,38,39,36,43,44);(13,17,29,41,19,32,18,33)(14,22,34,43,24,39,23,40)(15,26,16,30,28,42,27,31)(20,35,37,44,21,36,25,38);(5,7,8,11)(6,10,9,12)(17,32)(22,39)(26,42)(30,31)(33,41)(35,36)(38,44)(40,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=hi,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,d^c=di,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374843);