Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1450 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)(17,18,21,27,22,25,37,26)(19,28,32,41,30,23,31,39)(20,33,36,40,35,42,34,24)(29,44,46,38,45,48,47,43);(2,5)(3,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,30)(18,23,25,28)(20,29,35,45)(21,32,37,31)(24,38,40,43)(26,41,27,39)(33,48,42,44)(34,47,36,46);(1,3,4,7)(2,6,5,8)(10,14)(15,16)(17,20,22,35)(18,24,25,40)(19,29,30,45)(21,34,37,36)(23,38,28,43)(26,33,27,42)(31,46,32,47)(39,44,41,48);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(18,25)(19,30)(24,40)(26,27)(29,45)(31,32)(33,42)(46,47);(9,12,13,11)(10,15,14,16)(17,21,22,37)(18,26,25,27)(19,31,30,32)(20,34,35,36)(23,39,28,41)(24,33,40,42)(29,46,45,47)(38,44,43,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=f,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6782527);