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) = C25×C4 |
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 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(11,15)(13,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(2,6)(7,8)(9,10,12,14)(11,13,15,16)(18,22)(19,26)(24,28)(27,29)(36,45)(39,44)(42,48)(46,47);(1,2,5,6)(3,7,4,8)(17,18)(19,23)(20,28)(21,22)(24,30)(25,26)(27,32)(29,31)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,26)(18,23,22,25)(20,29,30,27)(24,32,28,31)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47);(1,4,5,3)(2,8,6,7)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=fgh,d^2=gh,e^2=fh,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, 3301855);