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 | |
1314 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)(11,15)(13,16)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,47)(46,48);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(18,19)(20,27)(24,26)(28,32)(34,38)(36,45)(39,41)(42,46);(1,2,4,5)(3,6,7,8)(17,19,22,18)(20,23,27,25)(21,24,29,26)(28,31,32,30)(33,35,37,43)(34,39,38,41)(36,46,45,42)(40,48,44,47);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,45)(46,48);(1,3)(2,6)(4,7)(5,8)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,36,37,45)(34,40,38,44)(35,42,43,46)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,c^2=gh,d^2=fghi,e^2=g,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, 5208836);