Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(10,13)(11,14)(15,16)(17,18,21,23,20,22,19,24);(1,3,6,2)(4,7,5,8)(17,18,19,24,20,22,21,23);(1,4,6,5)(2,7,3,8)(9,10,14,16,12,15,11,13)(18,22)(23,24);(1,5,6,4)(2,7,3,8)(17,19,20,21)(18,23,22,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72343127, #72343147, #72343171 |
| maximal normal subgroup of | #72343127, #72343147, #72343171 |
| maximal quotient of | #72343565, #72343574 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126641);