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 | #24234 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1732 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)(4,8)(5,6)(9,10)(11,20)(12,24)(13,26)(14,30)(15,19)(16,23)(17,28)(18,27)(21,35)(22,32)(25,33)(29,39)(31,36)(34,40)(37,38);(2,6)(7,8)(9,11,15,23)(10,16,19,20)(12,14,22,27)(13,21,29,37)(17,33,34,31)(18,32,30,24)(25,28,36,40)(26,38,39,35);(1,3,5,4)(2,7,6,8)(9,12,14,23,15,22,27,11)(10,16,18,24,19,20,30,32)(13,25,28,37,29,36,40,21)(17,31,26,38,34,33,39,35);(1,3,5,4)(2,7,6,8)(9,13,27,40,15,29,14,28)(10,17,30,39,19,34,18,26)(11,21,22,36,23,37,12,25)(16,31,32,35,20,33,24,38);(1,4,5,3)(2,7,6,8)(9,14,15,27)(10,18,19,30)(11,22,23,12)(13,28,29,40)(16,32,20,24)(17,26,34,39)(21,36,37,25)(31,35,33,38)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,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, 7472634);