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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1739 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,13)(14,19)(15,33)(16,24)(17,29)(18,32)(20,27)(22,39)(23,26)(25,36)(28,31)(30,40)(34,38)(35,37);(2,5)(6,8)(9,11,14,24)(10,16,19,21)(12,26,28,40)(13,30,31,23)(15,34,27,25)(17,37,32,39)(18,35,29,22)(20,38,33,36);(1,3,4,7)(2,6,5,8)(9,12,27,18,14,28,15,29)(10,17,33,31,19,32,20,13)(11,22,34,40,24,35,25,26)(16,30,38,39,21,23,36,37);(9,13,15,32,14,31,27,17)(10,18,20,12,19,29,33,28)(11,23,25,39,24,30,34,37)(16,35,36,40,21,22,38,26);(2,5)(6,8)(10,19)(12,28)(16,21)(18,29)(20,33)(22,35)(26,40)(36,38)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=g,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,d^c=di,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, 7467931);