Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#120 |
| prime radical | C2×2−1+4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
961 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3,5,6,8,7)(9,10,14,12)(11,17,20,24)(13,21,23,16)(15,22,18,19);(9,11)(10,15)(12,18)(13,22)(14,20)(16,24)(17,21)(19,23);(9,12,14,10)(11,18,20,15)(13,16,23,21)(17,22,24,19);(1,3,4,6,5,7,8,2)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=eh,f^b=fh,e^c=eh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 115955);