Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 | |
1897 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)(4,5)(6,8)(9,10)(11,20)(12,24)(13,18)(14,27)(15,19)(16,21)(17,25)(22,30)(23,31)(26,34)(28,32)(29,33)(35,37)(36,39)(38,40);(2,5)(6,7)(11,21)(13,14)(16,20)(17,33)(18,27)(22,37)(23,28)(25,29)(30,35)(31,32)(36,38)(39,40);(9,11,15,23)(10,16,19,32)(12,22,26,38)(13,21,14,28)(17,30,33,40)(18,31,27,20)(24,35,34,39)(25,36,29,37);(9,12,15,26)(10,17,19,33)(11,22,23,38)(13,25,14,29)(16,30,32,40)(18,34,27,24)(20,35,31,39)(21,36,28,37);(1,3,4,8)(2,6,5,7)(9,13,15,14)(10,18,19,27)(11,21,23,28)(12,25,26,29)(16,31,32,20)(17,34,33,24)(22,36,38,37)(30,39,40,35)Polycyclic presentation:
2^9|c^2=i,d^2=i,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,e^b=ei,g^b=gi,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, 7425775);