Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | |
1317 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)(11,15)(12,13)(14,16)(17,18)(19,28)(20,31)(21,36)(22,26)(23,39)(24,30)(25,35)(27,38)(29,45)(32,42)(33,40)(34,48)(37,44)(41,47)(43,46);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,26,28)(20,32,34,33)(21,23,37,38)(25,41,43,45)(27,44,39,36)(29,46,47,35)(31,40,48,42);(10,13)(14,15)(17,20,23,35,22,34,38,46)(18,25,27,31,26,43,39,48)(19,29,21,32,30,47,37,33)(24,40,36,41,28,42,44,45);(9,11,12,16)(10,14,13,15)(17,21,23,30,22,37,38,19)(18,24,27,36,26,28,39,44)(20,33,35,29,34,32,46,47)(25,42,31,45,43,40,48,41);(1,4,5,3)(2,7,8,6)(18,26)(20,34)(24,28)(27,39)(29,47)(32,33)(35,46)(36,44)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=h,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504382);