Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,13,12)(11,14,16,15)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(9,10,13,12)(11,15,16,14)(18,22)(20,29)(23,25)(26,30)(34,36)(37,45)(39,41)(42,48);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,29)(30,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,6)(4,8)(5,7)(17,19,21,27)(18,23,22,25)(20,30,29,26)(24,32,28,31)(33,36,38,34)(35,39,43,41)(37,40,45,44)(42,47,48,46);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,20,21,29)(18,24,22,28)(19,26,27,30)(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=fh,b^2=fgh,c^2=ghi,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3165380);