Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10,12,14)(11,13,16,15)(17,18,20,23,21,19,22,24);(1,2)(3,7)(4,8)(5,6)(17,19,20,24,21,18,22,23);(1,3,5,4)(2,7,6,8)(10,13)(11,16)(14,15);(1,4,5,3)(2,7,6,8)(10,14)(13,15)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=e,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,f^c=fh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 137686);