Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #239383 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,21)(13,23)(14,17)(15,19)(18,22)(20,24);(1,3)(2,5)(4,7)(6,8)(9,11,14,20)(10,15,17,21)(12,22,19,13)(16,18,24,23);(5,8)(9,12,13,11,14,19,22,20)(10,16,18,21,17,24,23,15);(10,17)(15,21)(16,24)(18,23)Polycyclic presentation:
2^9|b^2=h,c^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72310676, #72310679, #72310688, … |
| maximal normal subgroup of | #72310676, #72310679, #72310688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146580);