Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(9,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(30,34)(32,40)(35,37)(38,44);(15,22)(18,21)(23,28)(26,27)(29,30,33,34)(31,37,39,35)(32,36,40,41)(38,43,44,42);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,30)(31,35)(32,41)(33,34)(36,40)(37,39)(38,43)(42,44);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)(29,31,33,39)(30,35,34,37)(32,38,40,44)(36,42,41,43);(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,32)(30,36)(31,38)(33,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=fgi,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4304517);