Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,8)(9,12)(14,18)(15,21)(16,24)(26,28)(29,30,33,36)(31,37,39,34)(32,35,40,42)(38,43,44,41);(5,6,10,8)(7,9,11,12)(14,18)(16,24)(19,22)(23,27)(32,40)(35,42)(38,44)(41,43);(5,7,10,11)(6,9,8,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,31,33,39)(30,34,36,37)(32,38,40,44)(35,41,42,43);(1,2)(3,4)(5,8,10,6)(7,9,11,12)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(5,9)(6,11)(7,8)(10,12)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,32,33,40)(30,35,36,42)(31,38,39,44)(34,41,37,43)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fgh,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5526929);