Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,25)(13,18)(14,20)(16,27)(17,23)(19,30)(21,34)(22,32)(24,33)(26,31)(28,35)(29,36)(37,39)(38,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,26)(13,22)(14,24)(16,34)(17,19)(18,32)(20,33)(21,27)(23,30)(25,31)(28,37)(29,38)(35,39)(36,40);(9,12,14,28)(10,16,20,36)(11,21,24,38)(13,29,30,27)(15,31,33,39)(17,34,32,40)(18,25,19,35)(22,26,23,37);(9,11)(10,17)(12,21)(13,22)(14,24)(15,18)(16,34)(19,33)(20,32)(23,30)(25,31)(26,29)(27,37)(28,38)(35,39)(36,40);(2,6)(5,7)(9,13,14,30)(10,18,20,19)(11,22,24,23)(12,27,28,29)(15,32,33,17)(16,35,36,25)(21,37,38,26)(31,34,39,40)Polycyclic presentation:
2^9|b^2=f,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421535);