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 | |
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:
(1,2,4,5)(3,6,7,8)(9,10,14,19)(11,22,25,16)(12,27,29,37)(13,18,26,24)(15,21,31,20)(17,32,36,30)(23,38,39,34)(28,33,40,35);(2,5)(6,8)(9,11)(10,16)(12,28)(13,15)(14,25)(17,34)(18,20)(19,22)(21,24)(23,30)(26,31)(27,35)(29,40)(32,39)(33,37)(36,38);(9,12,14,29)(10,17,19,36)(11,23,25,39)(13,28,26,40)(15,32,31,30)(16,33,22,35)(18,38,24,34)(20,37,21,27);(1,3,4,7)(2,6,5,8)(9,13,14,26)(10,18,19,24)(11,15,25,31)(12,23,29,39)(16,20,22,21)(17,35,36,33)(27,38,37,34)(28,30,40,32);(9,10)(11,24)(12,17)(13,16)(14,19)(15,20)(18,25)(21,31)(22,26)(23,34)(27,30)(28,33)(29,36)(32,37)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=f,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, 6483296);