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 | |
5309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,10,13)(7,14,12,9)(8,16,18,11)(15,20,17,19);(5,7,9,6,10,12,14,13)(8,17,19,11,18,15,20,16);(1,2,3,4)(5,8,10,18)(6,11,13,16)(7,15,12,17)(9,20,14,19);(7,12)(9,14)(15,17)(19,20);(8,18)(11,16)(15,17)(19,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,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 | #72219649, #72219652, #72219673, … |
| maximal normal subgroup of | #72219649, #72219652, #72219673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350530);