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 | |
1748 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)(11,23,27,17)(12,29,32,38)(13,22,16,19)(14,20,30,40)(18,26,37,31)(24,36,33,39)(25,34,28,35);(1,2)(3,6)(4,8)(5,7)(9,11,15,27)(10,17,21,23)(12,30,32,14)(13,25,16,28)(18,36,37,39)(19,34,22,35)(20,38,40,29)(24,31,33,26);(9,12,13,31,15,32,16,26)(10,18,22,38,21,37,19,29)(11,24,28,14,27,33,25,30)(17,20,34,39,23,40,35,36);(1,3,5,4)(2,6,7,8)(9,13,15,16)(10,19,21,22)(11,25,27,28)(12,31,32,26)(14,24,30,33)(17,34,23,35)(18,29,37,38)(20,39,40,36);(1,4,5,3)(2,6,7,8)(9,14,13,33,15,30,16,24)(10,20,22,39,21,40,19,36)(11,26,28,12,27,31,25,32)(17,18,34,38,23,37,35,29)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,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, 6718454);