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 | |
2070 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)(4,8)(5,6)(9,10)(11,16)(12,23)(13,25)(14,30)(15,18)(17,28)(19,34)(20,31)(21,29)(22,33)(24,32)(26,27)(35,36)(37,38)(39,40);(2,6)(7,8)(9,11,15,22)(10,16,18,33)(12,24,27,39)(13,29,30,20)(14,31,25,21)(17,34,35,37)(19,36,38,28)(23,32,26,40);(1,3,5,4)(2,7,6,8)(9,12,25,36,15,27,14,28)(10,17,30,26,18,35,13,23)(11,19,31,39,22,38,21,24)(16,32,29,37,33,40,20,34);(1,4,5,3)(2,8,6,7)(9,13,25,10,15,30,14,18)(11,20,31,16,22,29,21,33)(12,26,36,35,27,23,28,17)(19,37,39,40,38,34,24,32);(2,6)(7,8)(12,27)(17,35)(19,38)(23,26)(24,39)(28,36)(32,40)(34,37)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fgi,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, 7469025);