Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1300 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)(3,5,7,8)(11,15)(14,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,27)(28,31)(30,32)(33,34,39,36)(35,42,40,37)(38,44,47,41)(43,45,46,48);(10,13)(14,16)(18,19)(20,27)(21,29)(30,31)(33,35,39,40)(34,37,36,42)(38,46,47,43)(41,48,44,45);(9,10,12,13)(11,14,15,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,36,39,34)(35,37,40,42)(38,44,47,41)(43,45,46,48);(1,2)(3,5)(4,6)(7,8)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37)(34,40)(35,36)(38,45)(39,42)(41,46)(43,44)(47,48);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=h,d^2=g,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4458670);