Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6443052 |
| 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 | |
1716 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)(14,36)(15,20)(16,21)(17,25)(23,49)(24,50)(26,40)(27,38)(28,43)(29,44)(30,60)(31,45)(32,39)(33,62)(34,47)(35,48)(37,41)(42,52)(46,53)(51,70)(54,63)(55,64)(56,66)(57,68)(58,72)(59,69)(61,65)(67,71);(9,11,14,25,16,26,15,37)(10,17,20,22,21,41,36,40)(12,27,30,52,29,55,31,23)(13,32,33,53,35,61,34,24)(18,42,45,38,44,49,60,64)(19,46,47,39,48,50,62,65)(28,56,58,71,57,51,59,54)(43,67,69,66,68,63,72,70);(2,5)(6,8)(9,12,15,31,16,29,14,30)(10,18,20,45,21,44,36,60)(11,23,25,27,26,52,37,55)(13,28,34,59,35,57,33,58)(17,38,40,42,41,64,22,49)(19,43,47,69,48,68,62,72)(24,51,32,54,53,56,61,71)(39,63,46,66,65,67,50,70);(9,13,14,33,16,35,15,34)(10,19,36,62,21,48,20,47)(11,24,37,61,26,53,25,32)(12,28,30,58,29,57,31,59)(17,39,22,50,41,65,40,46)(18,43,60,72,44,68,45,69)(23,51,55,71,52,56,27,54)(38,63,49,70,64,67,42,66);(1,3,4,7)(2,6,5,8)(12,29)(18,44)(23,52)(27,55)(28,57)(30,31)(38,64)(42,49)(43,68)(45,60)(51,56)(54,71)(58,59)(63,67)(66,70)(69,72)Polycyclic presentation:
2^9|b^2=f,c^2=h,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,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375045);