Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#95 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3086 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)(3,7,8,5)(4,6)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,17,23)(12,20)(13,21)(14,19)(16,18,22,24);(3,8)(5,7)(11,19)(15,23)(18,24)(20,21);(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,18,23,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,b^a=be,e^a=eh,g^a=ghi,c^b=ch,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72234697, #72234800, #72234835, … |
| maximal normal subgroup of | #72234697, #72234800, #72234835, … |
| maximal quotient of | #72234704, #72281407, #72281549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88321);