Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499263 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13341 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
625 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 625 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(7,14)(10,18)(15,20)(16,19);(6,10)(7,15)(8,17)(11,16)(12,18)(13,19)(14,20);(2,4)(5,6)(7,16)(8,11)(9,12)(10,15)(13,17)(14,19)(18,20);(6,11,12,13)(7,14)(10,16,18,19)(15,20)Polycyclic presentation:
2^9|a^2=f,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gh,h^b=hi,d^c=dh,e^c=ei,g^c=gh,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72239803, #72239808, #72239836, … |
| maximal normal subgroup of | #72239803, #72239808, #72239836, … |
| maximal quotient of | #72250666 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406670);