Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #35737 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,31,40)(20,25,22,27)(21,33,36,26)(29,38,37,44)(30,46,47,39)(32,43,35,41)(34,48,45,42);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,31)(18,24,28,40)(20,29,22,37)(21,34,36,45)(25,38,27,44)(26,42,33,48)(30,35,47,32)(39,43,46,41);(10,14)(13,15)(17,20)(18,25)(21,35)(22,23)(26,43)(27,28)(29,37)(30,47)(32,36)(33,41)(38,44)(39,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,23,22)(18,25,28,27)(19,29,31,37)(21,32,36,35)(24,38,40,44)(26,41,33,43)(30,45,47,34)(39,48,46,42);(2,5)(6,8)(11,16)(13,15)(17,21,23,36)(18,26,28,33)(19,30,31,47)(20,32,22,35)(24,39,40,46)(25,41,27,43)(29,45,37,34)(38,48,44,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6606523);