Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1231 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)(3,7)(4,6)(5,8)(9,10,12,13)(11,14,15,16)(17,18)(19,25)(20,34)(21,27)(22,28)(23,29)(24,30)(26,31)(32,40)(33,41)(35,44)(36,43)(37,42)(38,45)(39,46)(47,48);(1,3,4,8)(2,5,6,7)(17,19,23,33)(18,25,29,41)(20,21,31,24)(22,37,38,35)(26,30,34,27)(28,42,45,44)(32,36,47,39)(40,43,48,46);(10,13)(14,16)(17,20,24,19,23,31,21,33)(18,26,30,41,29,34,27,25)(22,37,39,47,38,35,36,32)(28,44,46,40,45,42,43,48);(2,6)(5,7)(17,21,23,24)(18,27,29,30)(19,31,33,20)(22,36,38,39)(25,26,41,34)(28,43,45,46)(32,37,47,35)(40,42,48,44);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,22,24,39,23,38,21,36)(18,28,30,46,29,45,27,43)(19,32,31,37,33,47,20,35)(25,40,26,42,41,48,34,44)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=h,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=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6739628);