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) = 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 | |
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)(18,22)(19,25)(24,29)(26,31)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(2,5)(6,8)(9,10,13,14)(11,15,12,16)(18,22)(20,27)(23,28)(26,31)(36,45)(39,44)(42,48)(46,47);(9,11,13,12)(10,15,14,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,16,14,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,3)(2,6)(4,7)(5,8)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fgh,d^2=f,e^2=h,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, 6039718);