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 |
| prime radical | #3499 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 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)(11,17)(12,24)(13,22)(14,19)(15,29)(16,20)(18,27)(21,35)(23,30)(25,34)(26,39)(28,36)(31,33)(32,37)(38,40);(2,6)(7,8)(9,11,16,23)(10,17,20,30)(12,25,28,38)(13,29,31,19)(14,22,15,33)(18,35,37,26)(21,32,39,27)(24,34,36,40);(1,3,5,4)(2,7,6,8)(9,12,14,32,16,28,15,27)(10,18,29,36,20,37,19,24)(11,21,33,38,23,39,22,25)(13,26,30,40,31,35,17,34);(1,3,5,4)(2,7,6,8)(9,13,15,17,16,31,14,30)(10,11,19,22,20,23,29,33)(12,26,27,34,28,35,32,40)(18,21,24,25,37,39,36,38);(1,4,5,3)(2,7,6,8)(9,14,16,15)(10,19,20,29)(11,22,23,33)(12,27,28,32)(13,30,31,17)(18,36,37,24)(21,38,39,25)(26,34,35,40)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,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, 7472635);