Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574100 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | D4×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1893 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,20)(11,21,23,32)(12,26,28,34)(13,19,14,18)(16,25,31,24)(17,29,33,27)(22,36,37,30)(35,39,40,38);(1,3)(2,5)(4,7)(6,8)(9,11,14,25,15,23,13,24)(10,16,19,32,20,31,18,21)(12,22,29,39,28,37,27,38)(17,30,34,40,33,36,26,35);(3,7)(5,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,26)(19,34)(20,33)(21,35)(23,37)(24,38)(25,39)(31,36)(32,40);(1,3)(2,5)(4,7)(6,8)(11,23)(16,31)(21,32)(22,37)(24,25)(30,36)(35,40)(38,39);(3,7)(5,8)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6499074);