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 | |
1759 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,24)(13,26)(14,19)(15,31)(16,20)(18,29)(21,37)(22,33)(23,34)(25,35)(27,32)(28,36)(30,40)(38,39);(2,5)(6,8)(9,11)(10,17)(12,15)(13,21)(14,22)(16,23)(18,35)(19,33)(20,34)(24,31)(25,29)(26,37)(27,28)(30,38)(32,36)(39,40);(1,3,4,7)(2,6,5,8)(9,12,14,23,16,22,15,11)(10,17,31,33,20,34,19,24)(13,27,28,21,30,25,29,38)(18,35,40,37,36,32,26,39);(9,13,14,28,16,30,15,29)(10,18,31,40,20,36,19,26)(11,21,12,25,23,38,22,27)(17,32,33,39,34,35,24,37);(2,5)(6,8)(9,14,16,15)(10,19,20,31)(11,22,23,12)(13,28,30,29)(17,33,34,24)(18,26,36,40)(21,27,38,25)(32,39,35,37)Polycyclic presentation:
2^9|c^2=fgi,d^2=gi,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,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, 7467930);