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 |
| prime radical | #61195 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1762 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,25)(13,30)(14,19)(15,21)(17,28)(18,24)(20,32)(22,35)(23,34)(26,33)(27,36)(29,37)(31,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11,15,24)(10,16,21,18)(12,26,29,27)(13,14,23,32)(17,35,38,40)(19,34,20,30)(22,31,39,28)(25,33,37,36);(9,12,15,29)(10,17,21,38)(11,22,24,39)(13,27,23,26)(14,31,32,28)(16,33,18,36)(19,25,20,37)(30,35,34,40);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,14)(12,27)(15,23)(16,21)(17,36)(19,30)(20,34)(22,31)(24,32)(25,35)(26,29)(28,39)(33,38)(37,40);(2,7)(6,8)(9,14,15,32)(10,19,21,20)(11,23,24,13)(12,28,29,31)(16,34,18,30)(17,37,38,25)(22,27,39,26)(33,35,36,40)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=f,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, 7424829);