Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1323 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)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,35)(21,27)(22,28)(23,29)(24,30)(26,37)(31,47)(32,42)(33,43)(34,44)(36,46)(38,45)(39,41)(40,48);(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,36)(21,32)(22,37)(23,34)(24,40)(26,28)(27,42)(29,44)(30,48)(31,33)(35,46)(38,39)(41,45)(43,47);(2,5)(6,8)(10,14)(11,16)(17,20,24,33,23,37,21,38)(18,26,30,45,29,35,27,43)(19,31,32,36,34,39,40,22)(25,41,42,28,44,47,48,46);(10,14)(13,15)(17,21,23,24)(18,27,29,30)(19,32,34,40)(20,33,37,38)(22,39,36,31)(25,42,44,48)(26,45,35,43)(28,41,46,47);(1,3,4,7)(2,6,5,8)(17,22,24,31,23,36,21,39)(18,28,30,47,29,46,27,41)(19,33,32,37,34,38,40,20)(25,43,42,26,44,45,48,35)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=fh,d^2=i,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6739972);