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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2397 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1645 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)(11,15)(12,13)(14,16)(17,18)(19,24)(20,31)(21,37)(22,39)(23,28)(25,36)(26,30)(27,33)(29,42)(32,40)(34,44)(35,47)(38,41)(43,45)(46,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,28,26)(20,32,35,34)(21,22,38,33)(25,42,43,48)(27,37,39,41)(29,45,46,36)(31,40,47,44);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,33,45,23,35,22,36)(18,25,39,47,28,43,27,31)(19,29,38,34,30,46,21,32)(24,40,37,48,26,44,41,42);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,22,30,23,38,33,19)(18,26,27,37,28,24,39,41)(20,34,36,29,35,32,45,46)(25,40,31,42,43,44,47,48);(1,4,5,3)(2,7,6,8)(20,35)(25,43)(29,46)(31,47)(32,34)(36,45)(40,44)(42,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fhi,d^2=fg,e^2=f,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, 7505852);