Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1384 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)(3,7)(9,10,12,14)(11,13,15,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,46)(47,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(19,26)(21,29)(23,25)(24,28)(34,37)(39,41)(40,44)(46,47);(9,11)(10,13)(12,15)(14,16)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(1,2)(3,6)(4,5)(7,8)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,34)(35,39,43,41)(36,44,45,40)(42,46,48,47)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6043261);