Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442587 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3455 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1850 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,18)(13,19)(14,37)(15,20)(16,21)(17,25)(23,41)(24,45)(26,40)(27,48)(28,54)(29,42)(30,60)(31,43)(32,44)(33,61)(34,46)(35,62)(36,47)(38,50)(39,53)(49,64)(51,65)(52,69)(55,66)(56,71)(57,67)(58,72)(59,68)(63,70);(9,11,16,27)(10,17,21,40)(12,28,32,50)(13,33,34,53)(14,26,15,25)(18,41,44,65)(19,45,46,69)(20,48,37,22)(23,30,51,31)(24,35,52,36)(29,56,57,49)(38,60,54,43)(39,62,61,47)(42,66,67,63)(55,58,70,59)(64,72,71,68);(2,5)(6,8)(9,12,14,30,16,32,15,31)(10,18,37,60,21,44,20,43)(11,23,25,50,27,51,26,28)(13,29,35,58,34,57,36,59)(17,38,48,65,40,54,22,41)(19,42,62,72,46,67,47,68)(24,49,53,70,52,56,33,55)(39,63,69,71,61,66,45,64);(9,13,14,35,16,34,15,36)(10,19,37,62,21,46,20,47)(11,24,25,53,27,52,26,33)(12,29,30,58,32,57,31,59)(17,39,48,69,40,61,22,45)(18,42,60,72,44,67,43,68)(23,49,50,70,51,56,28,55)(38,63,65,71,54,66,41,64);(1,3,4,7)(2,6,5,8)(13,34)(19,46)(24,52)(29,57)(33,53)(35,36)(39,61)(42,67)(45,69)(47,62)(49,56)(55,70)(58,59)(63,66)(64,71)(68,72)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6374450);