Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,22,27)(19,29,33,24)(20,25,35,42)(21,26,23,28)(30,45,47,38)(31,46,48,39)(32,44,37,40)(34,41,36,43);(2,6)(7,8)(10,13)(11,16)(17,19,22,33)(18,24,27,29)(20,30,35,47)(21,32,23,37)(25,38,42,45)(26,40,28,44)(31,36,48,34)(39,43,46,41);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,30)(21,36)(22,35)(23,34)(24,38)(26,43)(27,42)(28,41)(29,45)(31,37)(32,48)(33,47)(39,44)(40,46);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20)(18,25)(19,31)(21,34)(22,35)(23,36)(24,39)(26,41)(27,42)(28,43)(29,46)(30,37)(32,47)(33,48)(38,44)(40,45);(1,4,5,3)(2,7,6,8)(10,13)(14,15)(17,21,22,23)(18,26,27,28)(19,32,33,37)(20,34,35,36)(24,40,29,44)(25,41,42,43)(30,31,47,48)(38,39,45,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7494446);