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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,8,7,4)(11,15)(13,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(36,43)(39,46)(41,47)(44,48);(2,6)(3,7)(20,28)(23,30)(26,32)(29,31)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(3,7)(4,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,13,14,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(1,4)(2,7)(3,6)(5,8)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,36,37,43)(34,39,40,46)(35,41,42,47)(38,44,45,48)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fgi,d^2=hi,e^2=h,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, 5243904);