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 | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,15,16,13)(17,18,21,23)(19,27,28,24)(20,25,32,35)(22,26,33,36)(29,41,43,37)(30,42,44,38)(31,40,34,39)(45,47,46,48);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(18,23)(19,28)(20,31)(22,33)(25,39)(29,45)(32,34)(35,40)(37,47)(38,42)(41,48)(43,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,21,28)(18,24,23,27)(20,29,32,43)(22,30,33,44)(25,37,35,41)(26,38,36,42)(31,46,34,45)(39,48,40,47);(10,14)(13,15)(17,20,21,32)(18,25,23,35)(19,29,28,43)(22,34,33,31)(24,37,27,41)(26,39,36,40)(30,45,44,46)(38,48,42,47);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(18,23)(19,28)(25,35)(26,36)(29,43)(30,44)(39,40)(45,46)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=fgi,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,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, 7492094);