Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3496 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4877 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)(5,6,8,12,9,11,15,18)(7,10,14,17,13,16,19,20);(1,2)(3,4)(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20);(6,11)(8,15)(10,16)(14,19);(2,4)(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20);(7,13)(10,16)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,h^c=hi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72312266, #72312281, #72312520, … |
| maximal normal subgroup of | #72312266, #72312281, #72312520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286227);