Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1849 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)(6,9)(10,12)(15,21)(16,23)(18,24)(19,26)(30,35)(32,36)(38,43)(40,41);(13,14,17,20)(15,18,21,24)(16,19,23,26)(22,25,27,28)(29,30)(31,36)(32,37)(33,38)(34,35)(39,41)(40,44)(42,43);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,24)(22,25)(23,26)(27,28)(29,31,34,37)(30,36,35,32)(33,39,42,44)(38,41,43,40);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,18,20,24)(16,22,23,27)(19,25,26,28)(29,32,34,36)(30,37,35,31)(33,41,42,40)(38,39,43,44);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,22)(17,23)(18,25)(20,26)(21,27)(24,28)(29,33)(30,38)(31,39)(32,40)(34,42)(35,43)(36,41)(37,44)Polycyclic presentation:
2^9|b^2=h,c^2=f,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4326687);