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 | |
1420 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)(3,7,8,5)(9,10,13,14)(11,15,16,12)(20,27)(24,29)(25,31)(30,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(1,2,4,6)(3,5,8,7)(11,16)(12,15)(18,22)(23,28)(24,29)(30,32)(34,38)(39,46)(40,43)(47,48);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,46)(43,45)(44,48);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35,37,42)(34,39,38,46)(36,44,45,41)(40,48,43,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,36,37,45)(34,40,38,43)(35,41,42,44)(39,47,46,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fgh,d^2=gh,e^2=f,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, 4779864);