Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10,13,18,14,17,12,15)(11,16,21,23,19,24,20,22);(1,2,3,6,4,5,7,8)(10,15,17,18)(11,19)(16,23,24,22)(20,21);(9,11,13,21,14,19,12,20)(10,16,18,23,17,24,15,22);(10,17)(15,18)(16,24)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=h,g^2=i,b^a=be,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72285957, #72322122, #72322165 |
| maximal normal subgroup of | #72285957, #72322122, #72322165 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90084);