Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1726 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)(13,14)(15,21)(16,22)(17,25)(18,19)(20,28)(23,24)(26,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,44)(40,42)(41,43);(6,10)(11,12)(14,19)(15,22)(20,25)(23,26)(29,30,33,36)(31,37,39,34)(32,35,42,40)(38,44,41,43);(5,6,9,10)(7,11,8,12)(13,15)(14,16)(17,26)(18,22)(19,21)(20,27)(23,28)(24,25)(29,30)(31,34)(32,40)(33,36)(35,42)(37,39)(38,44)(41,43);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,15)(17,27)(18,21)(19,22)(20,26)(23,25)(24,28)(29,31,33,39)(30,34,36,37)(32,41,42,38)(35,44,40,43);(5,8,9,7)(6,12,10,11)(13,17)(14,20)(15,23)(16,24)(18,28)(19,25)(21,27)(22,26)(29,32,33,42)(30,35,36,40)(31,38,39,41)(34,43,37,44)Polycyclic presentation:
2^9|b^2=fg,c^2=fh,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3269423);