Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10)(11,16)(12,24)(13,18)(14,19)(15,21)(17,27)(20,29)(22,33)(23,25)(26,38)(28,34)(30,35)(31,36)(32,37)(39,40);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,21)(13,17)(14,30)(15,24)(18,27)(19,35)(20,25)(22,31)(23,29)(26,34)(28,38)(32,39)(33,36)(37,40);(9,12,15,16)(10,17,21,23)(11,18,24,20)(13,25,29,27)(14,26,32,33)(19,30,37,39)(22,34,38,36)(28,40,31,35);(1,3,4,7)(2,6,5,8)(9,13,15,29)(10,18,21,20)(11,17,24,23)(12,25,16,27)(14,28,32,31)(19,34,37,36)(22,30,38,39)(26,40,33,35);(2,5)(6,8)(9,14)(10,19)(11,22)(12,26)(13,28)(15,32)(16,33)(17,30)(18,34)(20,36)(21,37)(23,39)(24,38)(25,40)(27,35)(29,31)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578842);