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 | |
1427 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)(3,7)(4,5)(6,8)(10,12)(11,15)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(34,38)(39,41)(40,44)(46,47);(1,2)(3,7)(4,5)(6,8)(10,12)(14,16)(20,28)(23,30)(26,32)(29,31)(33,34,37,38)(35,41,43,39)(36,44,45,40)(42,46,48,47);(2,5)(6,7)(9,10,13,12)(11,16,15,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,43)(42,47)(46,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,16,15,14)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=ghi,d^2=fi,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, 5238789);