Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1728 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)(7,11)(10,12)(14,18)(15,21)(20,25)(22,27)(29,30,34,39)(31,36,40,35)(32,41,43,37)(33,42,44,38);(6,9)(10,12)(15,21)(19,24)(22,27)(26,28)(29,31,34,40)(30,35,39,36)(32,38,43,42)(33,41,44,37);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31)(30,36)(32,42)(33,37)(34,40)(35,39)(38,43)(41,44);(1,2,3,4)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,43)(30,37,39,41)(31,38,40,42)(33,36,44,35);(5,7)(6,10)(8,11)(9,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,44)(30,38,39,42)(31,37,40,41)(32,36,43,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4406383);