Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11757 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
809 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,17)(12,16)(13,15)(14,18)(19,21)(20,22)(23,24);(2,6,8,7)(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23);(10,15)(11,18)(12,20)(21,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35017);