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:
(1,2,3,4)(7,8)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(30,31)(32,39)(33,41)(42,43);(6,10)(11,12)(14,18)(19,21)(20,24)(26,27)(29,30,34,31)(32,37,39,35)(33,36,41,38)(40,43,44,42);(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,27)(26,28)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28)(29,32,34,39)(30,35,31,37)(33,40,41,44)(36,42,38,43);(5,8,9,7)(6,12,10,11)(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=g,d^2=fh,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, 4521156);