Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #95118 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
| unambiguous semidirect product | C22⋊#1304 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C27 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C27 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
9182 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 9182 conjugacy classes of subgroups — too many to draw. Only its 266 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 — 3 generators on 20 points:
(1,2)(3,4)(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);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,f^b=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214549, #72218387, #72218846, … |
| maximal normal subgroup of | #72214549, #72218387, #72218846, … |
| maximal quotient of | #72214449, #72218198, #72218313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12900);