Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470028 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,22,33)(12,16,23,31)(13,25,28,17)(15,26,30,21)(20,36,37,35)(24,34,27,32)(29,39,40,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,26)(14,22)(16,29)(17,33)(18,30)(19,25)(21,28)(23,37)(24,39)(27,38)(31,40)(32,35)(34,36);(3,7)(5,8)(9,12,14,23)(10,16,18,31)(11,20,22,37)(13,27,28,24)(15,29,30,40)(17,34,25,32)(19,35,33,36)(21,39,26,38);(3,7)(5,8)(12,23)(16,31)(20,37)(24,27)(29,40)(32,34)(35,36)(38,39);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=i,c^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377227);