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 | #2361 |
| prime radical | C2×#472 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,4,3,6,8)(9,10,14,18,15,11,13,17)(12,19,22,24,23,16,21,20);(1,3,5,8,4,2,6,7)(9,11,15,10)(12,20,23,24)(13,17,14,18)(16,21,19,22);(2,3)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24);(9,12,13,21,15,23,14,22)(10,16,17,24,11,19,18,20)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,c^b=ci,d^b=de,e^b=eh,g^b=gh,f^c=fh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 136388);