Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1703 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)(7,10)(8,9)(11,12)(13,14,17,18)(15,21,23,19)(16,20,25,24)(22,27,28,26)(29,30,35,39)(31,37,32,36)(33,44,38,34)(40,41,42,43);(7,11)(10,12)(14,18)(19,21)(20,24)(26,27)(29,31,35,32)(30,36,39,37)(33,40,38,42)(34,41,44,43);(6,9)(10,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28)(29,32,35,31)(30,37,39,36)(33,40,38,42)(34,41,44,43);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,33,35,38)(30,34,39,44)(31,40,32,42)(36,41,37,43);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)(29,34)(30,38)(31,41)(32,43)(33,39)(35,44)(36,42)(37,40)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5382874);