Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,12,15,16)(18,22)(19,25)(24,29)(27,31)(33,34,39,36)(35,37,43,40)(38,44,46,41)(42,47,48,45);(10,14)(11,15)(20,28)(24,29)(27,31)(30,32)(33,35,39,43)(34,40,36,37)(38,42,46,48)(41,47,44,45);(11,15)(12,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,39,34)(35,40,43,37)(38,41,46,44)(42,45,48,47);(1,2)(3,5)(4,6)(7,8)(9,11,13,15)(10,12,14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,37,39,40)(34,35,36,43)(38,45,46,47)(41,42,44,48);(1,3,4,7)(2,5,6,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,38,39,46)(34,41,36,44)(35,42,43,48)(37,45,40,47)Polycyclic presentation:
2^9|a^2=fghi,b^2=h,c^2=h,d^2=f,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5071604);