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) = 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 | |
1946 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)(5,6,9,7)(8,11,12,10)(13,14)(15,16)(17,21)(18,19)(20,23)(22,24)(25,28)(26,27)(32,39)(35,42)(37,43)(40,44);(14,19)(16,23)(21,27)(24,28)(29,30,33,36)(31,34,38,41)(32,35,39,42)(37,40,43,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,18,20)(14,16,19,23)(17,25,26,22)(21,28,27,24)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(5,8)(6,10)(7,11)(9,12)(13,16)(14,20)(15,19)(17,24)(18,23)(21,25)(22,27)(26,28)(29,31,33,38)(30,34,36,41)(32,37,39,43)(35,40,42,44);(13,17)(14,21)(15,22)(16,24)(18,26)(19,27)(20,25)(23,28)(29,32)(30,35)(31,37)(33,39)(34,40)(36,42)(38,43)(41,44)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=i,d^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, 4295421);