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 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(10,14)(12,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(10,14)(11,15)(18,19)(23,25)(24,26)(30,31)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(9,10,13,14)(11,12,15,16)(17,19,22,18)(20,25,28,23)(21,24,29,26)(27,30,32,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,3,5,8)(2,6,4,7)(9,11,13,15)(10,12,14,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,35)(34,38)(36,41)(37,42)(39,44)(40,45)(43,47)(46,48);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,14)(13,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,43)(34,39,40,46)(35,41,42,47)(38,44,45,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgh,c^2=h,d^2=fhi,e^2=ghi,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, 4768071);