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×#3500 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,17,11,16,9,15)(7,12,19,27,20,28,23,31)(8,14,21,30,22,13,24,29)(18,26,33,35,34,25,32,36);(1,2)(3,4)(5,7,11,20)(6,12,16,28)(8,18,22,34)(9,23,10,19)(13,25,14,26)(15,31,17,27)(21,33,24,32)(29,36,30,35);(5,8)(6,13)(7,18)(9,21)(10,24)(11,22)(12,25)(14,16)(15,29)(17,30)(19,32)(20,34)(23,33)(26,28)(27,35)(31,36);(2,4)(5,8)(6,14)(7,18)(9,24)(10,21)(11,22)(12,26)(13,16)(15,29)(17,30)(19,33)(20,34)(23,32)(25,28)(27,35)(31,36);(5,9,11,10)(6,15,16,17)(7,19,20,23)(8,21,22,24)(12,27,28,31)(13,29,14,30)(18,32,34,33)(25,35,26,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,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 | #72311140, #72311147, #72311157, … |
| maximal normal subgroup of | #72311140, #72311147, #72311157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286230);