Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1788 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)(9,12)(13,14,18,15)(16,21,23,19)(17,22,25,20)(24,26,28,27)(29,30,34,32)(31,35,37,38)(33,40,41,36)(39,44,43,42);(5,6,8,10)(7,9,11,12)(14,15)(16,23)(20,22)(24,28)(31,37)(35,38)(39,43)(42,44);(1,2,3,4)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,26,28,27)(29,31,34,37)(30,35,32,38)(33,39,41,43)(36,42,40,44);(5,7,8,11)(6,9,10,12)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,32,34,30)(31,38,37,35)(33,40,41,36)(39,44,43,42);(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,33,34,41)(30,36,32,40)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=gh,d^2=fhi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4302620);