Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #386 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14,16,18,17,20,15,19);(6,9,11,10)(14,18)(15,16)(19,20);(2,4)(14,18)(15,16)(19,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72215791, #72216007, #72216008, … |
| maximal normal subgroup of | #72215791, #72216007, #72216008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13152);