Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(18,21)(19,23)(22,24);(2,4)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,24,23,21);(10,13)(14,16)(17,18)(19,24)(20,22)(21,23);(2,4)(9,11,12,15)(10,14,13,16)(17,18)(19,24)(20,22)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=gi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi,e^d=ei,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264155, #72264168, #72264320, … |
| maximal normal subgroup of | #72264155, #72264168, #72264320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270488);