Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,10)(8,12)(15,21)(16,23)(19,24)(20,25)(29,30,33,34)(31,37,38,35)(32,40,42,36)(39,43,41,44);(5,6,9,11)(7,8,10,12)(14,18)(19,24)(20,25)(26,28)(30,34)(31,38)(36,40)(39,41);(5,7,9,10)(6,8,11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,33,38)(30,35,34,37)(32,41,42,39)(36,44,40,43);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,42)(41,44);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,42)(30,36,34,40)(31,39,38,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4288023);