Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #180869 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#97 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
620 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)(3,6)(4,5)(7,8)(9,10,11,15,12,17,19,23,13,16,18,24,20,14,21,22);(2,5)(3,7)(10,14,16,17)(11,18)(15,23,24,22)(19,21);(1,3,4,7)(2,6,5,8)(9,11,12,19,13,18,20,21)(10,15,17,23,16,24,14,22);(2,5)(6,8)(10,16)(14,17)(15,24)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,e^2=i,f^2=h,h^2=i,b^a=be,d^a=deh,g^a=gi,c^b=ceh,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239316, #72239317, #72239325, … |
| maximal normal subgroup of | #72239316, #72239317, #72239325, … |
| maximal quotient of | #72309924, C2×#493286, #72317207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400482);