Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628867 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,14)(11,15)(16,21)(17,23)(18,20)(22,24);(2,5)(6,8)(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,24,23);(9,11)(10,15)(12,17)(13,18)(14,20)(16,22)(19,24)(21,23);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,24)(20,23);(1,3,4,7)(2,6,5,8)(12,19)(16,21)(17,24)(22,23)Polycyclic presentation:
2^9|b^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,g^c=gi,h^c=hi,e^d=ei,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257818, #72311985, #72312002, … |
| maximal normal subgroup of | #72257818, #72311985, #72312002, … |
| maximal quotient of | #72264186, #72264510, #72264524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536569);