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 | |
1320 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)(9,10,13,16)(11,12,15,14)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30)(34,38)(35,41)(36,44)(47,48);(9,10)(11,14)(12,15)(13,16)(20,28)(24,29)(26,32)(30,31)(33,34,37,38)(35,42,41,39)(36,40,44,46)(43,48,45,47);(9,11)(10,14)(12,16)(13,15)(17,19,22,27)(18,23,21,25)(20,26,28,32)(24,30,29,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,46)(45,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20,22,28)(18,24,21,29)(19,26,27,32)(23,30,25,31)(33,35,37,41)(34,39,38,42)(36,45,44,43)(40,48,46,47);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,16)(13,14)(17,21,22,18)(19,25,27,23)(20,29,28,24)(26,31,32,30)(33,36)(34,40)(35,43)(37,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=fg,d^2=fgh,e^2=fghi,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, 4819224);