Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,14)(8,13)(9,15)(10,16)(11,17)(12,19)(18,20);(5,7,10,19,11,8,9,18)(6,12,15,14,17,20,16,13);(1,2,3,4)(5,8)(6,13)(7,11)(9,19)(10,18)(12,15)(14,17)(16,20);(5,7)(6,14)(8,11)(9,18)(10,19)(12,16)(13,17)(15,20);(5,9,11,10)(6,15,17,16)(7,18,8,19)(12,14,20,13)Polycyclic presentation:
2^9|b^2=f,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312281, D4×#3018, #72312715, … |
| maximal normal subgroup of | #72312281, D4×#3018, #72312715, … |
| maximal quotient of | C2×#6443305 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350501);