Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C23 | C43 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
2474 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 2474 conjugacy classes of subgroups — too many to draw. Only its 282 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:
(5,6)(7,11,12,9)(8,10)(13,14)(15,19)(16,18)(17,20);(5,7)(6,9)(8,12)(10,11)(13,15)(14,17,18,19)(16,20);(1,2,3,4)(6,10)(9,11)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,b^a=bd,c^a=ch,d^a=dh,f^a=fi,d^b=di,e^b=eh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72267706 |
| maximal normal subgroup of | #72267706 |
| maximal quotient of | #72267709 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12526);