Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(11,15)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,38,36)(35,41,43,39)(37,44,46,40)(42,47,45,48);(1,2,4,6)(3,5,7,8)(18,22)(23,25)(24,28)(30,31)(34,36)(39,41)(40,44)(47,48);(1,2)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,36,38,34)(35,41,43,39)(37,44,46,40)(42,48,45,47);(1,3)(2,5)(4,7)(6,8)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=ghi,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4541982);