Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1604 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)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,30,35,39)(31,37,32,36)(33,44,38,34)(40,43,42,41);(5,6)(7,9)(8,10)(11,12)(14,18)(19,21)(20,24)(26,27)(29,31,35,32)(30,36,39,37)(33,42,38,40)(34,41,44,43);(7,12)(9,11)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,27)(26,28)(29,32,35,31)(30,37,39,36)(33,42,38,40)(34,41,44,43);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28)(29,33,35,38)(30,34,39,44)(31,40,32,42)(36,43,37,41);(5,7)(6,9)(8,12)(10,11)(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)(29,34)(30,38)(31,41)(32,43)(33,39)(35,44)(36,40)(37,42)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=g,d^2=fgi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4577253);