Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60835 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(3,7)(6,8)(9,10,15,20,16,21,14,37)(11,22,27,42,25,54,26,17)(12,28,31,46,32,45,39,18)(13,19,35,50,36,51,38,49)(23,40,56,69,29,43,60,52)(24,53,33,65,58,47,59,41)(30,61,64,48,34,66,63,44)(55,68,62,72,57,70,67,71);(2,5)(3,7)(9,11,15,27,16,25,14,26)(10,17,37,54,21,42,20,22)(12,29,39,56,32,23,31,60)(13,33,38,24,36,59,35,58)(18,43,28,52,46,40,45,69)(19,47,50,41,51,53,49,65)(30,62,64,57,34,67,63,55)(44,71,66,70,48,72,61,68);(1,2)(3,6)(4,5)(7,8)(9,12,16,32)(10,18,21,46)(11,23,25,29)(13,34,36,30)(14,31,15,39)(17,40,42,43)(19,48,51,44)(20,45,37,28)(22,52,54,69)(24,57,58,55)(26,56,27,60)(33,62,59,67)(35,64,38,63)(41,70,65,68)(47,71,53,72)(49,66,50,61);(2,5)(6,8)(9,13,16,36)(10,19,21,51)(11,24,25,58)(12,30,32,34)(14,38,15,35)(17,41,42,65)(18,44,46,48)(20,50,37,49)(22,53,54,47)(23,55,29,57)(26,59,27,33)(28,61,45,66)(31,64,39,63)(40,68,43,70)(52,71,69,72)(56,62,60,67);(1,3,4,7)(2,6,5,8)(11,25)(17,42)(22,54)(23,29)(24,58)(26,27)(33,59)(40,43)(41,65)(47,53)(52,69)(55,57)(56,60)(62,67)(68,70)(71,72)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7056618);