Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1359 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 1359 conjugacy classes of subgroups — too many to draw. Only its 194 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:
(2,3)(5,6)(7,15)(8,18)(9,20)(10,13)(11,14)(12,16)(17,19);(1,2,4,3)(7,12)(8,19)(15,16)(17,18);(5,7,16,13,9,17,19,20,10,12,15,6,11,18,8,14);(1,3,4,2)Polycyclic presentation:
2^9|b^2=e,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72224852, #72224877 |
| maximal normal subgroup of | #72224852, #72224877 |
| maximal quotient of | #72224844 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410314);