Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| 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 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2032 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)(3,7)(4,8)(5,6)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,24)(22,35)(23,31)(25,33)(26,27)(28,34)(29,32)(30,37)(36,40)(38,39);(2,6)(7,8)(9,10)(11,21)(12,26)(13,19)(14,18)(15,20)(16,23)(17,27)(22,30)(24,31)(25,33)(28,32)(29,34)(35,37)(36,39)(38,40);(1,3,5,4)(2,7,6,8)(9,11,13,23,15,25,14,24)(10,16,19,33,20,31,18,21)(12,22,27,36,29,38,28,37)(17,30,34,39,32,40,26,35);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,26)(19,34)(20,32)(21,35)(23,36)(24,37)(25,38)(31,40)(33,39);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,21,31,33)(17,26,32,34)(22,36,38,37)(30,35,40,39)Polycyclic presentation:
2^9|c^2=fgi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351322);