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) = C26×C4 |
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 | |
1979 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)(6,9)(10,12)(13,14)(15,21)(16,23)(17,25)(18,19)(20,27)(22,24)(26,28)(33,41)(36,40)(38,44)(42,43);(7,11)(10,12)(14,19)(16,21)(20,25)(24,28)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,26)(18,23)(19,21)(20,28)(22,27)(24,25)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,26)(27,28)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(13,17,18,27)(14,20,19,25)(15,22,23,26)(16,24,21,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=h,d^2=fgh,e^2=f,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, 4517477);