Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,20,24,16)(12,25,28,35)(13,19)(14,29)(15,18)(17,27,34,30)(21,33)(22,36,23,32)(26,31)(37,39)(38,40);(9,11)(10,16)(12,26)(13,23)(14,22)(15,24)(17,33)(18,20)(19,36)(21,27)(25,37)(28,39)(29,32)(30,38)(31,35)(34,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17,18,34)(11,21)(13,27)(14,30)(15,28)(16,31,20,37)(19,25,29,35)(22,26)(23,39)(24,38)(32,40,36,33)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fh,g^b=gh,f^c=fh,g^c=ghi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26706);