Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511785 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2826 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 2826 conjugacy classes of subgroups — too many to draw. Only its 188 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,19)(14,16)(15,17)(18,20);(6,10)(7,13)(11,18)(12,17)(14,20)(16,19);(7,13)(8,15)(11,19)(16,18);(2,4)(5,7,9,13)(6,11,12,19)(8,14,15,20)(10,16,17,18)Polycyclic presentation:
2^9|d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,d^b=di,f^b=fhi,d^c=di,e^c=eh,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72233522, #72234427, #72249427, … |
| maximal normal subgroup of | #72233522, #72234427, #72249427, … |
| maximal quotient of | #72233132, #72233368, #72234569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418983);