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 | |
1296 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)(3,8)(4,7)(5,6)(11,15)(13,16)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,31,30,32)(33,34)(35,38)(36,44)(37,41)(39,43)(40,46)(42,48)(45,47);(2,6)(3,4)(18,22)(20,29)(23,25)(26,30)(33,35,37,43)(34,38,41,39)(36,45,46,42)(40,48,44,47);(1,3,5,4)(2,7,6,8)(9,10,12,14)(11,13,15,16)(17,19,21,27)(18,23,22,25)(20,30,29,26)(24,32,28,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,48)(44,47)(45,46);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(1,2)(3,8)(4,7)(5,6)(17,20,21,29)(18,24,22,28)(19,26,27,30)(23,31,25,32)(33,36,37,46)(34,40,41,44)(35,42,43,45)(38,47,39,48)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fghi,d^2=fi,e^2=fh,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, 5238873);