Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2052 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,29)(13,19)(14,15)(16,21)(17,26)(18,32)(20,34)(23,39)(24,30)(25,27)(28,37)(31,33)(35,38)(36,40);(2,5)(6,8)(9,11,15,26)(10,17,14,22)(12,30,19,35)(13,24,29,38)(16,27,34,28)(18,36,31,39)(20,25,21,37)(23,33,40,32);(9,12,15,19)(10,18,14,31)(11,23,26,40)(13,21,29,20)(16,33,34,32)(17,24,22,38)(25,39,37,36)(27,30,28,35);(1,3,4,7)(2,6,5,8)(9,13,15,29)(10,19,14,12)(11,24,26,38)(16,18,34,31)(17,35,22,30)(20,33,21,32)(23,25,40,37)(27,36,28,39);(9,14,15,10)(11,25,26,37)(12,31,19,18)(13,33,29,32)(16,21,34,20)(17,27,22,28)(23,39,40,36)(24,30,38,35)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482653);