Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×C4×C4⋊C4 |
| prime radical | C24 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(10,14,16,19)(11,17)(15,22,23,21)(18,24);(1,2,3,6,4,5,7,8)(9,10)(11,15)(12,19)(13,16)(14,20)(17,23)(18,22)(21,24);(9,11,12,18,13,17,20,24)(10,15,19,22,16,23,14,21);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,e^2=h,g^2=i,b^a=be,c^a=ch,d^b=di,e^b=eh,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#183351, #72213016, #72216075, … |
| maximal normal subgroup of | C2×#183351, #72213016, #72216075, … |
| maximal quotient of | C2×#183351 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90547);