Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #25115 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1486 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)(11,13)(12,14)(15,16)(17,18,24,36,23,27,22,28)(19,29,21,25,32,42,31,39)(20,26,37,46,34,41,35,40)(30,43,33,38,44,48,45,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,25,27,39)(20,30,34,44)(21,24,31,22)(26,38,41,47)(28,42,36,29)(33,37,45,35)(40,48,46,43);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20,23,34)(18,26,27,41)(19,30,32,44)(21,33,31,45)(22,35,24,37)(25,38,39,47)(28,40,36,46)(29,43,42,48);(3,8)(5,7)(11,16)(13,15)(17,21,22,19,23,31,24,32)(18,25,28,29,27,39,36,42)(20,33,35,30,34,45,37,44)(26,38,40,43,41,47,46,48);(10,14)(13,15)(18,27)(20,34)(25,39)(28,36)(29,42)(30,44)(33,45)(35,37)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=hi,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6986775);