Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1454 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)(11,13)(12,14)(15,16)(17,18,23,27,24,28,22,38)(19,29,31,41,32,46,21,25)(20,26,35,44,36,45,39,43)(30,47,34,42,33,48,37,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,24,32)(18,25,28,41)(20,33,36,30)(21,22,31,23)(26,42,45,40)(27,46,38,29)(34,39,37,35)(43,48,44,47);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20,24,36)(18,26,28,45)(19,30,32,33)(21,37,31,34)(22,39,23,35)(25,40,41,42)(27,44,38,43)(29,47,46,48);(3,8)(5,7)(11,16)(13,15)(17,21,23,19,24,31,22,32)(18,25,27,29,28,41,38,46)(20,34,35,33,36,37,39,30)(26,42,44,48,45,40,43,47);(10,14)(13,15)(17,22,24,23)(18,27,28,38)(19,21,32,31)(20,35,36,39)(25,29,41,46)(26,43,45,44)(30,34,33,37)(40,48,42,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=fg,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6986628);