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 | |
1346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,4)(3,8)(9,10,13,11)(12,15,16,14)(18,22)(23,25)(24,29)(30,31)(33,34,38,39)(35,40,41,36)(37,42,46,47)(43,45,48,44);(1,2,5,4)(3,6,8,7)(20,28)(24,29)(26,32)(30,31)(34,39)(35,41)(42,47)(43,48);(1,3)(2,6)(4,7)(5,8)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35)(34,40)(36,39)(37,43)(38,41)(42,45)(44,47)(46,48);(1,4,5,2)(3,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,40)(34,41,39,35)(37,45,46,44)(42,43,47,48);(1,3)(2,7)(4,6)(5,8)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,46)(34,42,39,47)(35,43,41,48)(36,44,40,45)Polycyclic presentation:
2^9|a^2=fhi,b^2=fh,c^2=ghi,d^2=gh,e^2=fg,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, 5638454);