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:
(2,6)(3,4)(11,15)(13,16)(18,22)(19,25)(24,29)(27,31)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,2,5,6)(3,7,4,8)(20,28)(24,29)(27,31)(30,32)(35,41)(38,45)(42,47)(46,48);(1,3,5,4)(2,7,6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,2)(3,8)(4,7)(5,6)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fgi,d^2=fghi,e^2=fgh,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, 5243641);