Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,8,6,7,5,3)(9,10)(11,19,21,15)(12,16)(13,18)(14,17)(20,24,22,23);(1,3,5,7,6,8,4,2)(9,11,13,22,14,21,12,20)(10,15,18,24,17,19,16,23);(2,7)(3,8);(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,f^2=i,g^2=hi,b^a=be,c^a=ci,e^a=eh,c^b=ci,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72214391, #72214857, #72214874, … |
| maximal normal subgroup of | #72214391, #72214857, #72214874, … |
| maximal quotient of | C2×#181106, #72309128, #72309153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88302);