Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1362 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18,23,28)(19,29,31,26)(20,25,32,40)(21,24,35,39)(22,36,37,27)(30,43,44,41)(33,38,45,48)(34,47,46,42);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,21,23,31,37,35)(18,24,27,29,28,39,36,26)(20,30,34,33,32,44,46,45)(25,38,42,43,40,48,47,41);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20,23,32)(18,25,28,40)(19,30,31,44)(21,33,35,45)(22,34,37,46)(24,38,39,48)(26,41,29,43)(27,42,36,47);(2,6)(5,7)(11,16)(14,15)(19,31)(20,32)(21,35)(24,39)(25,40)(26,29)(34,46)(42,47);(2,6)(5,7)(17,21,22,31,23,35,37,19)(18,26,27,24,28,29,36,39)(20,33,34,44,32,45,46,30)(25,41,42,38,40,43,47,48)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=h,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7514265);