Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1382 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,14)(13,15)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(34,38)(35,41)(40,43)(42,47);(19,25)(22,29)(26,31)(30,32)(33,34,37,38)(35,39,41,45)(36,43,44,40)(42,48,47,46);(2,6)(3,8)(9,11,14,13)(10,15,12,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,3,5,8)(2,7,6,4)(9,12,14,10)(11,15,13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(1,4,5,7)(2,8,6,3)(9,13,14,11)(10,15,12,16)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=fi,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4877031);