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 | |
2053 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)(11,20)(12,26)(13,30)(14,18)(15,19)(16,23)(17,29)(21,25)(22,36)(24,33)(27,31)(28,34)(32,40)(35,39)(37,38);(2,5)(6,8)(9,10)(11,20)(12,27)(13,18)(14,30)(15,19)(16,24)(17,28)(21,25)(22,32)(23,33)(26,31)(29,34)(35,38)(36,40)(37,39);(9,11,15,25)(10,16,19,33)(12,22,29,39)(13,24,14,23)(17,32,26,37)(18,21,30,20)(27,35,34,36)(28,40,31,38);(1,3,4,7)(2,6,5,8)(9,10)(11,21)(12,27)(13,18)(14,30)(15,19)(16,23)(17,28)(20,25)(22,37)(24,33)(26,31)(29,34)(32,39)(35,40)(36,38);(9,12)(10,17)(11,22)(13,28)(14,31)(15,29)(16,32)(18,34)(19,26)(20,35)(21,36)(23,38)(24,40)(25,39)(27,30)(33,37)Polycyclic presentation:
2^9|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, 6482648);