Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1942 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)(6,9)(7,11)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30,35,32)(31,33,39,36)(34,37,43,40)(38,41,44,42);(7,11)(10,12)(15,21)(18,25)(22,27)(26,28)(29,31,35,39)(30,36,32,33)(34,38,43,44)(37,42,40,41);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)(29,32,35,30)(31,33,39,36)(34,40,43,37)(38,41,44,42);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,33)(30,31)(32,39)(34,42)(35,36)(37,44)(38,40)(41,43);(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,34)(30,37)(31,38)(32,40)(33,41)(35,43)(36,42)(39,44)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=f,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4772658);