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) = 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 | |
1764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,8)(7,10,11,12)(13,14)(15,21)(16,22)(17,25)(18,19)(20,28)(23,24)(26,27)(29,30,35,33)(31,36,32,39)(34,41,42,37)(38,44,40,43);(7,11)(10,12)(14,19)(15,22)(20,25)(23,26)(29,31,35,32)(30,36,33,39)(34,38,42,40)(37,43,41,44);(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,26)(18,22)(19,21)(20,27)(23,28)(24,25)(29,32,35,31)(30,36,33,39)(34,40,42,38)(37,43,41,44);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,16)(14,15)(17,27)(18,21)(19,22)(20,26)(23,25)(24,28)(29,33,35,30)(31,36,32,39)(34,41,42,37)(38,43,40,44);(13,17)(14,20)(15,23)(16,24)(18,28)(19,25)(21,27)(22,26)(29,34)(30,37)(31,38)(32,40)(33,41)(35,42)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fg,d^2=gh,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, 3249615);