Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4)(3,7,5,8)(9,10,13,11)(12,15,16,14)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,32)(30,31)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,3,6,5)(2,7,4,8)(9,10,13,11)(12,14,16,15)(18,19)(21,29)(23,25)(27,30)(37,45)(40,44)(42,48)(46,47);(1,4,6,2)(3,7,5,8)(9,11,13,10)(12,15,16,14)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,32,30,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(9,12)(10,14)(11,15)(13,16)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,31)(27,29)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(1,5,6,3)(2,7,4,8)(17,21,22,29)(18,24,19,26)(20,27,28,30)(23,31,25,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fghi,b^2=hi,c^2=ghi,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5712220);