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 | #24174 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1668 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)(11,20)(12,24)(13,26)(14,30)(15,19)(16,23)(17,28)(18,31)(21,36)(22,33)(25,35)(27,38)(29,40)(32,37)(34,39);(2,6)(7,8)(9,11,15,23)(10,16,19,20)(12,14,22,31)(13,27,29,21)(17,32,34,35)(18,33,30,24)(25,39,37,28)(26,36,40,38);(1,3,5,4)(2,7,6,8)(9,12,14,23,15,22,31,11)(10,16,18,24,19,20,30,33)(13,25,28,27,29,37,39,21)(17,32,26,38,34,35,40,36);(9,13,14,28,15,29,31,39)(10,17,18,26,19,34,30,40)(11,21,12,25,23,27,22,37)(16,32,24,38,20,35,33,36);(1,4,5,3)(2,7,6,8)(9,14,15,31)(10,18,19,30)(11,22,23,12)(13,28,29,39)(16,33,20,24)(17,26,34,40)(21,37,27,25)(32,36,35,38)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=g,e^2=f,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, 7471402);