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 | |
1362 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)(11,15)(13,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34)(35,41)(36,38)(37,40)(39,42)(43,48)(44,46)(45,47);(9,10,12,14)(11,13,15,16)(18,19)(23,25)(24,26)(30,31)(34,36)(35,42)(40,44)(43,45);(9,10)(11,13)(12,14)(15,16)(17,19,22,18)(20,25,28,23)(21,24,29,26)(27,30,32,31)(33,35)(34,39)(36,41)(37,45)(38,42)(40,48)(43,46)(44,47);(1,2)(3,6)(4,5)(7,8)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36,38,34)(35,41,42,39)(37,44,46,40)(43,48,45,47);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,13,14,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,46)(34,40,36,44)(35,43,42,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fg,d^2=f,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, 4656523);