Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437999 |
| 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 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5953 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)(7,10)(8,11)(9,12)(13,15)(14,18)(16,17)(19,20);(5,6)(7,13,15,10)(8,16,17,11)(9,12)(14,18)(19,20);(1,2,3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20);(5,8)(6,11)(7,14)(9,17)(10,18)(12,16)(13,19)(15,20);(7,15)(10,13)(14,20)(18,19)Polycyclic presentation:
2^9|b^2=f,c^2=g,b^a=bf,c^a=cg,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 | #72210507, #72225623, #72225634, … |
| maximal normal subgroup of | #72210507, #72225623, #72225634, … |
| maximal quotient of | #72240570, #72241086, #72249353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350549);