Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| 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 | C2×C4×C22⋊C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,4,3,6,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23);(2,3)(7,8)(9,11,13,21,14,20,12,19)(10,15,16,23,17,18,22,24);(1,3,5,8,4,2,6,7);(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,f^2=i,g^2=h,b^a=be,g^a=gh,c^b=ci,e^b=eh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72342988, #72342998, #72343004, … |
| maximal normal subgroup of | #72342988, #72342998, #72343004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89302);