Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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:
(3,7)(6,8)(9,10,13,11)(12,15,16,14)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,48)(45,46);(2,5)(6,8)(9,11,13,10)(12,15,16,14)(20,28)(23,30)(26,32)(29,31)(34,38)(35,41)(40,46)(43,44);(1,2,4,5)(3,6,7,8)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35,37,41)(34,39,38,42)(36,44,45,43)(40,48,46,47);(9,12)(10,14)(11,15)(13,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,47)(45,46);(1,3,4,7)(2,6,5,8)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,36,37,45)(34,40,38,46)(35,43,41,44)(39,47,42,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=h,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5150363);