Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2366 |
| prime radical | C2×#496 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,7,5,3,4,8)(9,10,15,19,16,11,14,18)(12,21,22,20,23,24,13,17);(1,3,6,8,5,2,4,7)(9,11,16,10)(12,17,23,20)(13,21,22,24)(14,18,15,19);(1,4,5,6)(2,7,3,8)(9,12,14,13,16,23,15,22)(10,17,19,21,11,20,18,24);(9,13,14,23,16,22,15,12)(10,17,18,24,11,20,19,21)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=dh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138675);