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)(19,30)(20,36)(21,29)(22,27)(23,28)(24,34)(25,38)(26,40)(31,41)(32,42)(33,45)(35,47)(37,44)(39,43)(46,48);(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,37)(21,32)(22,38)(23,34)(25,44)(26,42)(27,36)(28,30)(29,47)(31,39)(33,48)(35,40)(41,45)(43,46);(2,5)(6,8)(10,14)(11,16)(17,20,21,39,23,38,40,33)(18,25,29,45,28,36,26,43)(19,31,35,37,34,48,32,22)(24,41,42,44,30,46,47,27);(10,14)(13,15)(17,21,23,40)(18,26,28,29)(19,32,34,35)(20,33,38,39)(22,31,37,48)(24,42,30,47)(25,45,36,43)(27,46,44,41);(1,3,4,7)(2,6,5,8)(17,22,40,48,23,37,21,31)(18,27,26,46,28,44,29,41)(19,33,32,38,34,39,35,20)(24,43,47,36,30,45,42,25)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=fhi,d^2=i,e^2=gh,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, 6739966);