Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1318 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)(6,8)(11,15)(14,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,47)(43,44)(46,48);(1,2,5,4)(3,6,7,8)(10,13)(14,16)(19,25)(22,28)(26,31)(29,32)(34,39)(35,41)(42,45)(43,48);(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35,38,41)(34,40,39,36)(37,43,47,48)(42,46,45,44);(1,3)(2,6)(4,8)(5,7)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32)(33,36,38,40)(34,41,39,35)(37,46,47,44)(42,43,45,48);(1,4,5,2)(3,8,7,6)(9,11)(10,14)(12,15)(13,16)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,37,38,47)(34,42,39,45)(35,43,41,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4820357);