Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(14,18)(15,21)(16,23)(26,28)(29,30,35,38)(31,33,32,36)(34,37,43,44)(39,41,40,42);(2,4)(15,21)(19,24)(22,27)(26,28)(29,31,35,32)(30,36,38,33)(34,39,43,40)(37,42,44,41);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,32,35,31)(30,36,38,33)(34,39,43,40)(37,41,44,42);(5,7,9,11)(6,10,8,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,33,35,36)(30,31,38,32)(34,42,43,41)(37,40,44,39);(5,8,9,6)(7,12,11,10)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,34,35,43)(30,37,38,44)(31,39,32,40)(33,41,36,42)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3699820);