Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1707 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)(6,8)(9,12)(14,18)(15,21)(20,25)(23,27)(29,30)(31,32)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43);(5,6,10,8)(7,9,11,12)(16,24)(20,25)(23,27)(26,28)(30,35)(31,37)(36,43)(38,44);(5,7,10,11)(6,9,8,12)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,31,34,37)(30,32,35,40)(33,38,42,44)(36,39,43,41);(1,2)(3,4)(5,8,10,6)(7,9,11,12)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,32)(30,31)(33,41)(34,40)(35,37)(36,44)(38,43)(39,42);(5,9)(6,11)(7,8)(10,12)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,33)(30,36)(31,38)(32,39)(34,42)(35,43)(37,44)(40,41)Polycyclic presentation:
2^9|b^2=fh,c^2=fg,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5524081);