Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533135 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4050 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)(3,7,8,5)(4,6);(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21);(1,3)(2,5)(4,8)(6,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24);(12,20)(15,22)(18,24)(21,23)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,b^a=bf,c^a=cg,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72339532, #72339539, #72339552 |
| maximal normal subgroup of | #72339532, #72339539, #72339552 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479763);