Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1786 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)(6,9)(7,11)(13,14)(15,21)(16,22)(17,25)(18,19)(20,27)(23,24)(26,28)(29,30,35,32)(31,37,33,36)(34,38,42,40)(39,44,41,43);(14,19)(15,22)(20,25)(23,28)(29,31,35,33)(30,36,32,37)(34,39,42,41)(38,43,40,44);(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,23)(18,22)(19,21)(20,24)(25,26)(27,28)(29,32,35,30)(31,37,33,36)(34,40,42,38)(39,44,41,43);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,26)(18,21)(19,22)(20,28)(23,25)(24,27)(29,33,35,31)(30,37,32,36)(34,39,42,41)(38,43,40,44);(13,17,18,27)(14,20,19,25)(15,23,22,28)(16,24,21,26)(29,34)(30,38)(31,39)(32,40)(33,41)(35,42)(36,43)(37,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=fg,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4342537);