Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12310 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#3292 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
856 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 856 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 20 points:
(1,2)(3,4)(5,6,10,15,13,16,19,14,11,12,9,20,8,7,17,18);(2,4)(6,12)(8,13)(9,19)(10,17)(14,15)(18,20);(5,7)(6,13)(8,12)(9,14)(10,18)(11,16)(15,19)(17,20)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=gi,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dh,e^b=eg,f^b=fghi,g^b=gi,d^c=dg,e^c=eg,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#12310, #72216412, #72219618, … |
| maximal normal subgroup of | C4×#12310, #72216412, #72219618, … |
| maximal quotient of | #72219610, #72219632, #72221506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39028);