Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C2×#509 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,3,4,6,8,7)(9,10,11,17,13,15,19,18)(12,21,20,16,23,14,22,24);(2,6)(3,7)(10,14)(11,19)(12,22)(15,21)(16,17)(18,24)(20,23);(1,3,5,6,4,7,8,2)(9,11,13,19)(12,22,23,20)(14,21)(16,24);(10,15)(14,21)(16,24)(17,18)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225324, #72286241, #72286259, … |
| maximal normal subgroup of | #72225324, #72286241, #72286259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391033);