Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1265 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(6,8)(9,10,16,21)(11,17,24,40)(12,27,30,60)(13,19,15,36)(14,32,35,61)(18,42,45,29)(20,31,47,34)(22,48,50,70)(23,39,26,57)(25,54,53,71)(28,59,37,46)(33,43,62,44)(38,63,65,51)(41,52,67,56)(49,69,58,68)(55,64,72,66);(1,2)(3,6)(4,5)(7,8)(9,11,16,24)(10,17,21,40)(12,22,30,50)(13,23,15,26)(14,25,35,53)(18,38,45,65)(19,39,36,57)(20,41,47,67)(27,48,60,70)(28,49,37,58)(29,51,42,63)(31,52,34,56)(32,54,61,71)(33,55,62,72)(43,64,44,66)(46,68,59,69);(9,12)(10,18)(11,22)(13,28)(14,33)(15,37)(16,30)(17,38)(19,43)(20,46)(21,45)(23,49)(24,50)(25,55)(26,58)(27,32)(29,34)(31,42)(35,62)(36,44)(39,64)(40,65)(41,68)(47,59)(48,54)(51,56)(52,63)(53,72)(57,66)(60,61)(67,69)(70,71);(2,5)(6,8)(9,13,16,15)(10,19,21,36)(11,23,24,26)(12,28,30,37)(14,31,35,34)(17,39,40,57)(18,43,45,44)(20,32,47,61)(22,49,50,58)(25,52,53,56)(27,59,60,46)(29,33,42,62)(38,64,65,66)(41,54,67,71)(48,69,70,68)(51,55,63,72);(1,3,4,7)(2,6,5,8)(11,24)(17,40)(22,50)(23,26)(25,53)(38,65)(39,57)(41,67)(48,70)(49,58)(51,63)(52,56)(54,71)(55,72)(64,66)(68,69)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531220);