Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18)(19,30)(20,33)(21,39)(22,40)(23,29)(24,27)(25,32)(26,37)(28,41)(31,42)(34,43)(35,44)(36,45)(38,48)(46,47);(1,3)(2,5)(4,8)(6,7)(17,19,23,32)(18,25,29,30)(20,34,37,36)(21,22,24,41)(26,43,33,45)(27,40,39,28)(31,35,47,38)(42,48,46,44);(10,13)(14,16)(17,20,21,35,23,37,24,38)(18,26,39,48,29,33,27,44)(19,31,41,36,32,47,22,34)(25,42,40,45,30,46,28,43);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,27,29,39)(19,22,32,41)(20,35,37,38)(25,40,30,28)(26,44,33,48)(31,34,47,36)(42,45,46,43);(3,8)(5,7)(17,22,21,19,23,41,24,32)(18,28,39,25,29,40,27,30)(20,36,35,47,37,34,38,31)(26,45,48,46,33,43,44,42)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=hi,d^2=g,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,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, 6729198);