Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438880 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,10)(7,13)(8,15)(11,18)(12,17)(16,19);(1,2)(3,4)(5,6,8,10,9,12,15,17)(7,11,14,16,13,19,20,18);(2,4)(5,7,8,14,9,13,15,20)(6,11,10,16,12,19,17,18);(2,4);(6,12)(10,17)(11,19)(16,18)Polycyclic presentation:
2^9|b^2=f,c^2=f,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213480, #72213527, #72213751, … |
| maximal normal subgroup of | #72213480, #72213527, #72213751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346076);