Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(13,14,17,18)(15,19,21,26)(16,23,25,20)(22,28,24,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(6,9)(10,12)(14,18)(15,21)(20,23)(22,24)(29,30,33,36)(31,34,39,37)(32,35,41,44)(38,42,40,43);(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,26)(22,27)(23,25)(24,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,43)(41,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15,17,21)(14,19,18,26)(16,24,25,22)(20,28,23,27)(29,31)(30,34)(32,40)(33,39)(35,43)(36,37)(38,41)(42,44);(13,16,17,25)(14,20,18,23)(15,22,21,24)(19,27,26,28)(29,32,33,41)(30,35,36,44)(31,38,39,40)(34,42,37,43)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=fh,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2518807);