Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12707 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
2476 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 2476 conjugacy classes of subgroups — too many to draw. Only its 202 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:
(5,6)(7,13,15,19)(8,16,17,11)(9,12)(10,14)(18,20);(1,2,3,4)(5,7)(6,10,12,18)(8,14)(9,15)(11,13,16,19)(17,20);(8,17)(11,16)(13,19)(14,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,e^b=ei,f^b=fhi,d^c=di,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72218057, #72218104, #72218121, … |
| maximal normal subgroup of | #72218057, #72218104, #72218121, … |
| maximal quotient of | #72218498, #72234075, #72235854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53675);