Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| 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 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,15,21)(11,22,26,17)(12,27,29,38)(13,30,16,19)(14,20,32,40)(18,28,35,25)(23,39,31,37)(24,34,33,36);(1,2)(3,6)(4,8)(5,7)(9,11)(10,17)(12,14)(13,24)(15,26)(16,33)(18,37)(19,34)(20,38)(21,22)(23,25)(27,40)(28,31)(29,32)(30,36)(35,39);(9,12,16,25,15,29,13,28)(10,18,19,38,21,35,30,27)(11,23,24,14,26,31,33,32)(17,20,36,37,22,40,34,39);(1,3,5,4)(2,6,7,8)(9,13,15,16)(10,19,21,30)(11,24,26,33)(12,28,29,25)(14,31,32,23)(17,34,22,36)(18,38,35,27)(20,39,40,37);(1,4,5,3)(2,6,7,8)(9,14,13,23,15,32,16,31)(10,20,30,39,21,40,19,37)(11,25,33,12,26,28,24,29)(17,35,34,38,22,18,36,27)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=f,e^2=fh,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,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718451);