Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6,9,10)(7,11,12,8)(16,23)(20,25)(21,27)(26,28)(31,37)(34,40)(38,43)(41,44);(5,7)(6,8)(9,12)(10,11)(14,18)(19,24)(20,25)(26,28)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(5,8,9,11)(6,7,10,12)(13,14,17,18)(15,19,22,24)(16,20,23,25)(21,26,27,28)(29,31)(30,34)(32,38)(33,37)(35,41)(36,40)(39,43)(42,44);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(5,8)(6,7)(9,11)(10,12)(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,27)(24,28)(29,32,33,39)(30,35,36,42)(31,38,37,43)(34,41,40,44)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4517124);