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 | |
1308 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)(3,7)(9,10,13,11)(12,15,16,14)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34)(35,39)(36,38)(37,40)(41,47)(42,43)(44,46)(45,48);(2,4)(6,8)(9,11,13,10)(12,15,16,14)(18,19)(20,27)(24,26)(28,32)(34,36)(39,43)(40,44)(47,48);(1,2,5,4)(3,8,7,6)(17,19,22,18)(20,23,27,25)(21,24,29,26)(28,31,32,30)(33,35,38,42)(34,39,36,43)(37,45,46,41)(40,48,44,47);(1,3,5,7)(2,6,4,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36,38,34)(35,39,42,43)(37,44,46,40)(41,47,45,48);(1,4,5,2)(3,8,7,6)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37,38,46)(34,40,36,44)(35,41,42,45)(39,47,43,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,c^2=fg,d^2=fh,e^2=fg,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, 4976314);