Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1736 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)(5,6)(7,8)(9,10)(11,12)(14,18)(15,21)(20,25)(23,27)(29,30,35,32)(31,36,33,39)(34,40,42,37)(38,44,41,43);(6,10)(7,12)(16,24)(20,25)(23,27)(26,28)(29,31,35,33)(30,36,32,39)(34,38,42,41)(37,43,40,44);(7,12)(8,11)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,32,35,30)(31,36,33,39)(34,40,42,37)(38,43,41,44);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28)(29,33,35,31)(30,36,32,39)(34,41,42,38)(37,43,40,44);(5,8,9,11)(6,7,10,12)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)(29,34)(30,37)(31,38)(32,40)(33,41)(35,42)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4989746);