Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1783 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)(7,9)(8,10)(11,12)(13,14,18,21)(15,17,22,19)(16,24,27,20)(23,28,25,26)(29,30,35,38)(31,34,32,36)(33,41,43,37)(39,44,40,42);(8,12)(10,11)(15,22)(17,19)(23,25)(26,28)(29,31,35,32)(30,36,38,34)(33,40,43,39)(37,42,41,44);(6,7)(10,11)(13,15)(14,19)(16,25)(17,21)(18,22)(20,26)(23,27)(24,28)(29,32,35,31)(30,34,38,36)(33,40,43,39)(37,42,41,44);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,27)(14,20,21,24)(15,23,22,25)(17,26,19,28)(29,33,35,43)(30,37,38,41)(31,39,32,40)(34,42,36,44);(5,8)(6,10)(7,11)(9,12)(13,17,18,19)(14,15,21,22)(16,26,27,28)(20,23,24,25)(29,34)(30,32)(31,38)(33,42)(35,36)(37,40)(39,41)(43,44)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=f,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4990078);