Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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:
(2,4)(5,6,8,10)(7,11,12,9)(14,18)(15,21)(20,25)(22,27)(29,30,35,38)(31,36,32,33)(34,37,42,44)(39,43,40,41);(5,6)(7,11)(8,10)(9,12)(15,21)(19,24)(22,27)(26,28)(29,31,35,32)(30,33,38,36)(34,40,42,39)(37,43,44,41);(5,6,8,10)(7,9,12,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,32,35,31)(30,36,38,33)(34,40,42,39)(37,43,44,41);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,33,35,36)(30,32,38,31)(34,41,42,43)(37,40,44,39);(5,7)(6,9)(8,12)(10,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,34,35,42)(30,37,38,44)(31,39,32,40)(33,41,36,43)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=gh,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4408135);