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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1758 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)(7,9)(8,10)(11,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(30,32)(31,37)(36,40)(39,43);(8,12)(10,11)(15,22)(16,25)(18,21)(19,24)(29,30,34,32)(31,38,37,35)(33,40,41,36)(39,42,43,44);(6,7)(10,11)(13,15,17,22)(14,18,20,21)(16,23,25,28)(19,26,24,27)(29,31,34,37)(30,35,32,38)(33,39,41,43)(36,42,40,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,21)(16,19)(17,20)(18,22)(23,27)(24,25)(26,28)(29,32,34,30)(31,38,37,35)(33,36,41,40)(39,42,43,44);(5,8,9,12)(6,10,7,11)(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,33,34,41)(30,36,32,40)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=h,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, 5527022);