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 | |
1703 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)(7,10)(8,9)(11,12)(13,14,17,18)(15,21,23,19)(16,24,25,20)(22,26,28,27)(29,30,35,32)(31,37,33,36)(34,38,42,40)(39,44,41,43);(7,11)(10,12)(14,18)(19,21)(20,24)(26,27)(29,31,35,33)(30,36,32,37)(34,39,42,41)(38,43,40,44);(6,9)(10,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28)(29,32,35,30)(31,36,33,37)(34,40,42,38)(39,43,41,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)(29,33,35,31)(30,37,32,36)(34,39,42,41)(38,43,40,44);(5,7,8,11)(6,10,9,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,27)(26,28)(29,34,35,42)(30,38,32,40)(31,39,33,41)(36,43,37,44)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,d^2=f,e^2=fg,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, 5519612);