Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,18)(15,21)(16,24)(26,28)(30,31)(32,39)(33,41)(42,43);(14,18)(19,22)(20,25)(26,28)(29,30,34,31)(32,35,39,37)(33,38,41,36)(40,43,44,42);(6,7)(10,11)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,31,34,30)(32,37,39,35)(33,38,41,36)(40,43,44,42);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28)(29,32)(30,35)(31,37)(33,40)(34,39)(36,42)(38,43)(41,44);(5,8,9,12)(6,10,7,11)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=fh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4851125);