Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18866 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,7)(3,8)(4,5)(10,14,16,19)(15,21,22,24);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,19)(13,16)(14,20)(17,22)(18,24)(21,23);(1,3,6,2)(4,8,5,7)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23);(1,4,6,5)(2,8,3,7)(11,17)(15,22)(18,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh,f^c=fh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93736);