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 | |
1718 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)(7,8)(11,12)(13,14)(15,18)(16,21)(17,20)(19,23)(22,25)(24,27)(26,28)(29,30)(31,34)(32,35)(33,36)(37,41)(38,42)(39,40)(43,44);(6,10)(11,12)(14,15)(16,23)(20,22)(24,28)(29,30,33,36)(31,34,38,42)(32,35,40,39)(37,41,44,43);(5,6)(7,11)(8,12)(9,10)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,26,28,27)(29,30)(31,34)(32,39)(33,36)(35,40)(37,43)(38,42)(41,44);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28)(29,31,33,38)(30,34,36,42)(32,37,40,44)(35,41,39,43);(5,8,9,7)(6,12,10,11)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,32)(30,35)(31,37)(33,40)(34,41)(36,39)(38,44)(42,43)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=g,d^2=gh,e^2=fgh,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, 4521935);