Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500450 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37418 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18,23,29)(19,30,33,25)(20,24,35,40)(21,26,38,43)(22,27,31,28)(32,45,47,41)(34,39,46,48)(36,42,37,44);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,31,35,23,33,22,20)(18,24,27,25,29,40,28,30)(21,32,36,46,38,47,37,34)(26,39,44,41,43,48,42,45);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,33,23,35,31,19)(18,25,28,24,29,30,27,40)(21,34,37,47,38,46,36,32)(26,41,42,39,43,45,44,48);(11,16)(13,15)(17,21,23,38)(18,26,29,43)(19,32,33,47)(20,34,35,46)(22,37,31,36)(24,39,40,48)(25,41,30,45)(27,44,28,42);(10,14)(13,15)(17,22,23,31)(18,27,29,28)(19,20,33,35)(21,36,38,37)(24,25,40,30)(26,42,43,44)(32,46,47,34)(39,45,48,41)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413800);