Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1349 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)(9,10,12,14)(11,13,15,16)(17,18)(19,29)(20,25)(21,24)(22,38)(23,28)(26,32)(27,30)(31,42)(33,41)(34,47)(35,48)(36,39)(37,40)(43,45)(44,46);(3,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19,30,37,23,32,22,21)(18,24,38,26,28,40,27,29)(20,31,45,34,33,46,35,36)(25,39,48,44,41,47,43,42);(1,3,4,8)(2,5,6,7)(17,20,23,33)(18,25,28,41)(19,31,32,46)(21,36,37,34)(22,35,30,45)(24,39,40,47)(26,44,29,42)(27,43,38,48);(2,6)(5,7)(11,15)(13,16)(19,32)(20,33)(21,37)(24,40)(25,41)(26,29)(35,45)(43,48);(17,21,22,32,23,37,30,19)(18,26,27,24,28,29,38,40)(20,34,35,31,33,36,45,46)(25,42,43,47,41,44,48,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=gi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7497829);