Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437995 |
| 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 | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5535 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)(7,10)(8,11)(9,12)(13,17)(14,18)(15,20)(16,19);(5,7)(6,10)(8,15,16,13)(9,14)(11,17,19,20)(12,18);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,19)(14,15)(18,20);(5,8)(6,11)(7,13)(9,16)(10,17)(12,19)(14,15)(18,20);(6,12)(10,18)(11,19)(17,20)Polycyclic presentation:
2^9|b^2=f,c^2=g,b^a=bf,c^a=cg,e^a=ei,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72260883, #72311822, #72311830, … |
| maximal normal subgroup of | #72260883, #72311822, #72311830, … |
| maximal quotient of | #72261094 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350545);