Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,14)(15,16)(20,27)(24,29)(25,31)(30,32)(34,38)(35,41)(36,44)(47,48);(10,14)(11,16)(18,22)(23,28)(24,29)(30,32)(33,34,37,38)(35,42,41,39)(36,45,44,40)(43,47,46,48);(11,16)(12,15)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,35,37,41)(34,39,38,42)(36,46,44,43)(40,48,45,47);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,14)(13,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48);(1,3,4,7)(2,5,6,8)(9,12,13,15)(10,11,14,16)(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)(33,36,37,44)(34,40,38,45)(35,43,41,46)(39,47,42,48)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=h,d^2=gh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 5093388);