Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,15,16,13)(20,31)(25,41)(28,43)(32,42)(33,35)(37,47)(44,48)(45,46);(1,3,5,4)(2,6,8,7)(11,16)(13,15)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,33)(28,44)(29,38)(30,40)(31,42)(34,39)(35,41)(37,46)(43,48)(45,47);(9,11,12,16)(10,13,14,15)(17,19,23,30)(18,24,27,40)(20,28,31,43)(21,34,22,29)(25,37,41,47)(26,39,36,38)(32,44,42,48)(33,46,35,45);(10,14)(13,15)(17,20,22,33,23,31,21,35)(18,25,26,32,27,41,36,42)(19,28,29,46,30,43,34,45)(24,37,39,44,40,47,38,48);(1,4,5,3)(2,7,8,6)(17,21,23,22)(18,26,27,36)(19,29,30,34)(20,33,31,35)(24,38,40,39)(25,42,41,32)(28,45,43,46)(37,44,47,48)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=gi,d^2=h,e^2=fi,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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7522583);