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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1813 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)(5,6,10,11)(7,12,8,9)(13,14)(15,21)(16,18)(17,24)(19,23)(20,26)(22,27)(25,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(6,11)(7,8)(14,16)(19,21)(20,24)(27,28)(29,30,33,36)(31,34,39,37)(32,35,41,40)(38,42,44,43);(5,7,10,8)(6,9,11,12)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27)(29,30)(31,34)(32,40)(33,36)(35,41)(37,39)(38,43)(42,44);(1,2,3,4)(5,8,10,7)(6,9,11,12)(13,16,18,14)(15,21,23,19)(17,24,26,20)(22,28,25,27)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,9)(6,7)(8,11)(10,12)(13,17,18,26)(14,20,16,24)(15,22,23,25)(19,27,21,28)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=g,d^2=hi,e^2=fh,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, 4579087);