Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6443050 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,22)(12,17)(13,18)(14,20)(15,21)(16,25)(19,29)(23,39)(24,44)(26,38)(27,48)(28,54)(30,40)(31,42)(32,43)(33,61)(34,45)(35,47)(36,50)(37,53)(41,56)(46,57)(49,63)(51,64)(52,69)(55,65)(58,72)(59,66)(60,68)(62,71)(67,70);(2,5)(6,7)(9,11,14,25,15,27,29,26)(10,16,19,22,21,38,20,48)(12,28,31,23,32,50,56,51)(13,33,35,24,34,53,57,52)(17,39,41,54,43,64,42,36)(18,44,46,61,45,69,47,37)(30,58,60,55,59,49,70,71)(40,65,67,72,66,62,68,63);(1,3,4,8)(2,6,5,7)(9,12,29,56,15,32,14,31)(10,17,19,41,21,43,20,42)(11,23,25,50,27,51,26,28)(13,30,57,70,34,59,35,60)(16,36,48,64,38,54,22,39)(18,40,46,67,45,66,47,68)(24,49,53,71,52,58,33,55)(37,62,69,72,61,65,44,63);(9,13,14,35,15,34,29,57)(10,18,20,47,21,45,19,46)(11,24,26,33,27,52,25,53)(12,30,31,60,32,59,56,70)(16,37,22,44,38,61,48,69)(17,40,42,68,43,66,41,67)(23,49,28,55,51,58,50,71)(36,62,39,63,54,65,64,72);(13,34)(18,45)(24,52)(30,59)(33,53)(35,57)(37,61)(40,66)(44,69)(46,47)(49,58)(55,71)(60,70)(62,65)(63,72)(67,68)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=ghi,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374080);