Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10,13,14)(11,12,16,15)(17,18)(19,23)(20,22)(21,28)(24,29)(25,31)(26,27)(30,32)(33,34,38,37)(35,39,41,43)(36,44,45,40)(42,48,47,46);(2,5)(6,8)(9,10,13,14)(11,15,16,12)(18,20)(21,29)(23,27)(25,32)(35,41)(36,45)(39,43)(40,44);(10,14)(11,16)(17,19,22,26)(18,23,20,27)(21,25,29,32)(24,30,28,31)(33,35,38,41)(34,39,37,43)(36,42,45,47)(40,46,44,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20,22,18)(19,23,26,27)(21,28,29,24)(25,30,32,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,24)(19,25)(20,28)(22,29)(23,30)(26,32)(27,31)(33,37,38,34)(35,43,41,39)(36,44,45,40)(42,48,47,46)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=fgh,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5885047);