Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628872 |
| 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 |
| prime radical | #35527 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(11,17)(12,19)(14,21)(16,23)(18,20)(22,24);(2,5)(6,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,22)(19,23)(20,24);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,20)(13,18)(15,22)(16,24)(17,19)(21,23);(12,19)(16,23)(17,20)(21,24);(10,15)(14,22)(16,23)(21,24)Polycyclic presentation:
2^9|c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^b=ei,g^c=gi,h^c=hi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72251973, #72251976, #72252048, … |
| maximal normal subgroup of | #72251973, #72251976, #72252048, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536372);