Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | |
2040 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)(11,23)(12,27)(13,22)(14,20)(15,21)(16,19)(17,25)(18,29)(24,35)(26,37)(28,34)(30,36)(31,39)(32,40)(33,38);(2,5)(3,7)(9,11,15,25)(10,17,21,23)(12,28,30,32)(13,24,16,26)(14,31,33,29)(18,38,39,20)(19,35,22,37)(27,40,36,34);(1,3,4,7)(2,6,5,8)(9,12,15,30)(10,18,21,39)(11,14,25,33)(13,29,16,31)(17,34,23,40)(19,27,22,36)(20,35,38,37)(24,32,26,28);(2,5)(6,8)(9,13,15,16)(10,19,21,22)(11,24,25,26)(12,29,30,31)(14,32,33,28)(17,35,23,37)(18,27,39,36)(20,34,38,40);(9,14,15,33)(10,20,21,38)(11,12,25,30)(13,28,16,32)(17,36,23,27)(18,35,39,37)(19,40,22,34)(24,31,26,29)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=i,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,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6576467);