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 | |
1450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,4)(5,8)(6,7)(9,10,13,14)(11,12,16,15)(18,22)(19,25)(24,29)(26,31)(33,34)(35,41)(36,44)(37,40)(38,43)(39,46)(42,47)(45,48);(1,3)(2,6)(4,8)(5,7)(10,14)(12,15)(19,25)(23,28)(26,31)(30,32)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(1,4)(2,7)(3,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,46)(45,48);(1,4)(2,3)(5,6)(7,8)(9,12,13,15)(10,11,14,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=h,c^2=fgh,d^2=i,e^2=fg,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, 4802746);