Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246503 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1720 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 172 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 — 4 generators on 20 points:
(2,3)(5,6)(7,9)(8,11)(10,12);(1,2,4,3)(7,12)(9,10)(14,17)(15,19)(18,20);(6,9)(7,12)(10,11)(13,14)(15,20)(16,18)(17,19);(1,3,4,2)(14,18)(17,20)Polycyclic presentation:
2^9|b^2=e,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72264142, #72264304, #72264340, … |
| maximal normal subgroup of | #72264142, #72264304, #72264340, … |
| maximal quotient of | #72279577 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 169783);