Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 | |
1822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,13,15)(11,19,22,31)(12,23,24,39)(14,27,26,18)(16,30,28,20)(17,33,29,25)(21,36,38,32)(34,37,40,35);(2,6)(7,8)(10,15)(11,20)(12,24)(14,26)(16,31)(17,29)(19,28)(21,37)(22,30)(32,40)(34,36)(35,38);(1,3,5,4)(2,7,6,8)(9,11,14,20,13,22,26,30)(10,16,18,19,15,28,27,31)(12,21,25,35,24,38,33,37)(17,32,23,34,29,36,39,40);(9,12,14,25,13,24,26,33)(10,17,18,23,15,29,27,39)(11,21,20,35,22,38,30,37)(16,32,19,34,28,36,31,40);(1,4,5,3)(2,7,6,8)(10,15)(11,22)(17,29)(18,27)(20,30)(21,38)(23,39)(35,37)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=g,e^2=fi,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,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472011);