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 | |
1714 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)(15,19)(16,18)(17,20)(21,22)(23,27)(24,26)(25,28)(30,32)(31,37)(33,41)(43,44);(6,10)(11,12)(14,16)(15,21)(20,24)(23,25)(29,30,34,32)(31,35,37,38)(33,36,41,40)(39,43,42,44);(5,6,9,10)(7,11,8,12)(13,15,18,21)(14,19,16,22)(17,25,26,23)(20,28,24,27)(29,31)(30,35)(32,38)(33,42)(34,37)(36,44)(39,41)(40,43);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,18,14)(15,22,21,19)(17,20,26,24)(23,27,25,28)(29,32,34,30)(31,38,37,35)(33,36,41,40)(39,43,42,44);(5,8,9,7)(6,12,10,11)(13,17,18,26)(14,20,16,24)(15,23,21,25)(19,27,22,28)(29,33,34,41)(30,36,32,40)(31,39,37,42)(35,43,38,44)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=fghi,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, 3386289);