Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196326 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #34980 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,21,24,23)(18,22);(2,5)(6,8)(9,11)(10,14)(12,17)(13,18)(15,23)(16,22)(19,21)(20,24);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(3,7)(6,8)Polycyclic presentation:
2^9|a^2=h,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225650, #72225654, #72225656, … |
| maximal normal subgroup of | #72225650, #72225654, #72225656, … |
| maximal quotient of | #72234623, #72237989, #72284794 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286929);