Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,28)(26,27)(29,30,33,34)(31,35,37,38)(32,36,40,42)(39,43,41,44);(2,4)(14,18)(16,25)(19,21)(22,26)(30,34)(31,37)(32,40)(43,44);(1,2,3,4)(5,6,9,8)(7,10,11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,25)(26,27)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,42)(41,43);(5,7,9,11)(6,10,8,12)(13,15)(14,19)(16,26)(17,23)(18,21)(20,28)(22,25)(24,27)(29,31)(30,35)(32,41)(33,37)(34,38)(36,44)(39,40)(42,43);(5,8,9,6)(7,12,11,10)(13,16,17,25)(14,20,18,24)(15,22,23,26)(19,27,21,28)(29,32)(30,36)(31,39)(33,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=g,c^2=fhi,d^2=i,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3211805);