Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196122 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,13,15,10)(8,12)(9,11)(14,20,17,18)(16,19);(1,2,4,3)(7,14)(8,16)(10,18)(12,19)(13,20)(15,17);(5,7)(6,10)(8,17)(9,15)(11,13)(12,20)(14,16)(18,19);(1,3,4,2)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#1904, #72252458, #72252468, … |
| maximal normal subgroup of | D4×#1904, #72252458, #72252468, … |
| maximal quotient of | #72264135, #72264303, C2×#360123 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267319);