Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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:
(2,5)(3,7)(10,14)(15,16)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,34,38,39)(35,42,40,36)(37,41,45,46)(43,48,47,44);(2,5)(6,8)(9,10,13,14)(11,15,12,16)(19,25)(23,26)(27,31)(30,32)(34,39)(35,40)(37,45)(44,48);(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35,38,40)(34,36,39,42)(37,43,45,47)(41,44,46,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,46)(45,48);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,16,14,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=f,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6039645);