Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470395 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,7,8,6)(11,15)(14,16)(17,18,22,27)(19,28,32,44)(20,25,33,24)(21,35,29,26)(23,36,39,31)(30,43,46,42)(34,40,45,41)(37,47,38,48);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,29,36,22,32,21,31)(18,23,35,44,27,39,26,28)(20,30,45,48,33,46,34,47)(24,37,41,43,25,38,40,42);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,20,22,33)(18,24,27,25)(19,30,32,46)(21,34,29,45)(23,37,39,38)(26,40,35,41)(28,42,44,43)(31,47,36,48);(10,13)(14,16)(17,20)(18,25)(19,30)(21,34)(22,33)(23,38)(24,27)(26,41)(28,43)(29,45)(31,47)(32,46)(35,40)(36,48)(37,39)(42,44);(9,11,12,15)(10,14,13,16)(20,33)(24,25)(30,46)(34,45)(37,38)(40,41)(42,43)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6402387);