Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437969 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,11,14)(7,15,16,20)(8,12,19,18)(9,13,10,17);(5,7,10,18,11,16,9,12)(6,8,13,15,14,19,17,20);(1,2,3,4)(7,16)(9,10)(13,17)(15,20);(5,8)(6,12)(7,17)(9,15)(10,20)(11,19)(13,16)(14,18);(5,9,11,10)(6,13,14,17)(7,12,16,18)(8,15,19,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72341753, #72341782, #72341788 |
| maximal normal subgroup of | #72341753, #72341782, #72341788 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350534);