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 | |
1364 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)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(9,11,14,12)(10,15,13,16)(18,22)(20,28)(23,25)(26,32)(35,41)(39,42)(43,47)(46,48);(1,2,4,6)(3,5,7,8)(9,12,14,11)(10,15,13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,3)(2,5)(4,7)(6,8)(9,13,14,10)(11,16,12,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=fhi,d^2=fgh,e^2=f,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, 4874685);