Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8610 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #380 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
894 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 894 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,11,12,9)(8,10)(14,19)(15,20)(16,17);(1,2,3,4)(13,14,16,19,18,20,17,15);(5,7)(6,9)(8,12)(10,11)(13,15,16,14,18,19,17,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,c^b=ch,f^c=fhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313650, #72313655, #72313663, … |
| maximal normal subgroup of | #72313650, #72313655, #72313663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15771);