Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(10,14)(11,16)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(9,10)(11,15)(12,16)(13,14)(20,28)(24,29)(26,32)(30,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(9,11)(10,15)(12,14)(13,16)(17,19,22,27)(18,23,21,25)(20,26,28,32)(24,30,29,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20,22,28)(18,24,21,29)(19,26,27,32)(23,30,25,31)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,14)(13,15)(17,21,22,18)(19,25,27,23)(20,29,28,24)(26,31,32,30)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fg,d^2=fgh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4819209);