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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(11,12)(15,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(34,38)(36,44)(39,41)(42,48);(2,5)(6,8)(10,14)(15,16)(19,26)(22,25)(27,32)(29,31)(33,34,37,38)(35,41,43,39)(36,45,44,40)(42,46,48,47);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(9,11,13,12)(10,15,14,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,25)(28,32)(30,31)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=ghi,d^2=gi,e^2=fgh,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, 4433511);