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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1706 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)(6,10)(8,11)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28)(31,37)(34,40)(38,43)(41,44);(6,10)(7,12)(14,15)(16,23)(20,22)(24,28)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(5,6,9,10)(7,11,12,8)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,26,28,27)(29,31)(30,34)(32,38)(33,37)(35,41)(36,40)(39,43)(42,44);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(5,8,9,11)(6,7,10,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(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=gh,c^2=h,d^2=f,e^2=fg,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, 4515572);