Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437798 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2548 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)(3,8)(4,6)(5,7)(9,10,14,20,16,22,15,21)(11,23,26,41,28,40,27,17)(12,18,32,45,31,44,36,49)(13,19,34,47,35,48,37,50)(24,42,29,56,52,63,54,38)(25,46,33,61,53,64,55,39)(30,43,59,67,58,66,60,68)(51,65,57,71,69,72,70,62);(9,11,16,28)(10,17,22,41)(12,29,31,54)(13,33,35,55)(14,27,15,26)(18,42,44,63)(19,46,48,64)(20,40,21,23)(24,36,52,32)(25,37,53,34)(30,51,58,69)(38,49,56,45)(39,50,61,47)(43,62,66,71)(57,59,70,60)(65,67,72,68);(1,3,5,4)(2,6,7,8)(9,12)(10,18)(11,24)(13,30)(14,32)(15,36)(16,31)(17,38)(19,43)(20,45)(21,49)(22,44)(23,42)(25,51)(26,29)(27,54)(28,52)(33,57)(34,59)(35,58)(37,60)(39,62)(40,63)(41,56)(46,65)(47,67)(48,66)(50,68)(53,69)(55,70)(61,71)(64,72);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,25)(12,30)(14,34)(15,37)(16,35)(17,39)(18,43)(20,47)(21,50)(22,48)(23,46)(24,51)(26,33)(27,55)(28,53)(29,57)(31,58)(32,59)(36,60)(38,62)(40,64)(41,61)(42,65)(44,66)(45,67)(49,68)(52,69)(54,70)(56,71)(63,72);(12,31)(18,44)(24,52)(29,54)(30,58)(32,36)(38,56)(42,63)(43,66)(45,49)(51,69)(57,70)(59,60)(62,71)(65,72)(67,68)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359229);