Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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:
(5,6,9,7)(8,11,12,10)(13,14)(15,21)(16,18)(17,24)(19,23)(20,25)(22,26)(27,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(2,4)(14,16)(19,21)(20,24)(26,27)(29,30,33,36)(31,34,39,37)(32,35,41,44)(38,42,40,43);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,43)(41,44);(5,8)(6,10)(7,11)(9,12)(13,16,18,14)(15,21,23,19)(17,20,25,24)(22,26,28,27)(29,31)(30,34)(32,40)(33,39)(35,43)(36,37)(38,41)(42,44);(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,32,33,41)(30,35,36,44)(31,38,39,40)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=fi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2540544);