Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,17)(12,26)(13,19)(14,31)(15,20)(16,21)(18,28)(22,40)(23,34)(24,36)(25,37)(27,33)(29,38)(30,39)(32,35);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,27)(13,28)(14,24)(15,32)(16,25)(18,19)(20,35)(21,37)(22,29)(23,30)(26,33)(31,36)(34,39)(38,40);(2,5)(6,7)(9,12,16,23)(10,18,21,38)(11,22,25,13)(14,29,15,28)(17,33,37,39)(19,36,40,35)(20,34,31,26)(24,30,32,27);(1,3,4,8)(2,6,5,7);(9,13,16,22)(10,19,21,40)(11,23,25,12)(14,30,15,27)(17,34,37,26)(18,36,38,35)(20,33,31,39)(24,29,32,28)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=f,e^2=i,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, 6578189);