Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 | |
1808 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)(3,5)(4,6)(7,8)(9,10,15,20,16,22,14,21)(11,23,25,17,27,40,26,35)(12,28,29,18,31,36,30,37)(13,19,33,38,24,34,32,39);(1,2)(3,7)(4,6)(5,8)(9,11,15,25,16,27,14,26)(10,17,21,23,22,35,20,40)(12,13,30,32,31,24,29,33)(18,19,36,38,37,34,28,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,24)(13,27)(14,29)(15,30)(16,31)(17,34)(19,35)(20,28)(21,36)(22,37)(23,39)(25,33)(26,32)(38,40);(3,8)(5,7)(9,13)(10,19)(11,12)(14,32)(15,33)(16,24)(17,18)(20,38)(21,39)(22,34)(23,28)(25,29)(26,30)(27,31)(35,37)(36,40);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,25,27,26)(12,29,31,30)(13,32,24,33)(17,23,35,40)(18,28,37,36)(19,38,34,39)Polycyclic presentation:
2^9|a^2=gi,b^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910258);