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 | |
1852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(29,30)(31,34)(32,37)(33,36)(35,39)(38,41)(40,43)(42,44);(5,7,8,11)(6,9,10,12)(15,22)(18,21)(23,28)(25,27)(30,31)(32,39)(36,38)(40,44);(5,7)(6,9)(8,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2)(3,4)(13,15,17,22)(14,18,20,21)(16,23,24,28)(19,25,26,27)(29,32)(30,35)(31,37)(33,40)(34,39)(36,42)(38,43)(41,44);(5,6)(7,9)(8,10)(11,12)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=f,d^2=fg,e^2=g,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, 4506186);