Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1728 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)(7,11)(13,14)(15,21)(16,23)(17,20)(18,19)(22,25)(24,28)(26,27)(30,34)(31,37)(32,40)(43,44);(7,11)(10,12)(14,19)(16,21)(20,27)(24,25)(29,30,33,34)(31,38,37,35)(32,36,40,42)(39,44,41,43);(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,22)(18,23)(19,21)(20,24)(25,27)(26,28)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,42)(41,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,25)(18,21)(19,23)(20,28)(22,27)(24,26)(29,31,33,37)(30,35,34,38)(32,41,40,39)(36,44,42,43);(13,17,18,26)(14,20,19,27)(15,22,23,28)(16,24,21,25)(29,32,33,40)(30,36,34,42)(31,39,37,41)(35,43,38,44)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=fg,d^2=fgh,e^2=f,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, 4775823);