Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439035 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2556 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)(9,10,16,21)(11,22,25,42)(12,27,30,46)(13,19,34,50)(14,35,15,20)(17,36,41,26)(18,37,45,31)(23,51,54,39)(24,47,55,65)(28,58,53,43)(29,57,60,68)(32,63,56,40)(33,49,38,48)(44,62,67,61)(52,69,71,64)(59,72,70,66);(2,5)(6,8)(9,11,14,26,16,25,15,36)(10,17,20,22,21,41,35,42)(12,28,31,23,30,53,37,54)(13,32,33,24,34,56,38,55)(18,43,27,51,45,58,46,39)(19,47,48,63,50,65,49,40)(29,52,61,70,60,71,62,59)(44,64,57,66,67,69,68,72);(1,3,4,7)(2,6,5,8)(9,12,14,31,16,30,15,37)(10,18,20,27,21,45,35,46)(11,23,26,53,25,54,36,28)(13,29,33,61,34,60,38,62)(17,39,22,43,41,51,42,58)(19,44,48,57,50,67,49,68)(24,52,56,70,55,71,32,59)(40,64,47,66,63,69,65,72);(9,13)(10,19)(11,24)(12,29)(14,33)(15,38)(16,34)(17,40)(18,44)(20,48)(21,50)(22,47)(23,52)(25,55)(26,56)(27,57)(28,59)(30,60)(31,61)(32,36)(35,49)(37,62)(39,64)(41,63)(42,65)(43,66)(45,67)(46,68)(51,69)(53,70)(54,71)(58,72);(11,25)(12,30)(17,41)(18,45)(22,42)(24,55)(26,36)(27,46)(29,60)(31,37)(32,56)(40,63)(44,67)(47,65)(57,68)(61,62)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6348045);