Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
3121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,11,8)(7,14,10,12)(9,17,18,20)(13,19,16,15);(1,2)(3,4)(6,12)(7,10)(8,14)(13,17)(15,19)(16,20);(2,4)(5,7,11,10)(6,12,8,14)(9,15,18,19)(13,17,16,20);(2,4)(5,8)(6,11)(7,14)(9,16)(10,12)(13,18)(15,20)(17,19);(5,9)(6,13)(7,15)(8,16)(10,19)(11,18)(12,17)(14,20)Polycyclic presentation:
2^9|a^2=i,c^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,f^b=fi,h^b=hi,h^d=hi,f^e=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421348);