Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1370 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)(18,22)(19,25)(20,27)(30,32)(33,34,38,39)(35,36,40,43)(37,41,47,48)(42,44,45,46);(9,10,13,12)(11,15,16,14)(19,25)(23,28)(26,31)(30,32)(34,39)(36,43)(41,48)(44,46);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,40)(34,36,39,43)(37,45,47,42)(41,46,48,44);(1,3,5,7)(2,6,4,8)(9,10,13,12)(11,14,16,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,40)(35,39)(37,46)(38,43)(41,42)(44,47)(45,48);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,47)(34,41,39,48)(35,42,40,45)(36,44,43,46)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fghi,d^2=ghi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3730814);