Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#453 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(5,6,8,7)(10,13)(11,15)(14,16);(1,2)(3,4)(5,7,8,6)(11,15)(13,16);(2,4)(9,10)(11,16)(12,14)(13,15);(11,15)(13,16)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216143, #72232809, #72233101, … |
| maximal normal subgroup of | #72216143, #72232809, #72233101, … |
| maximal quotient of | #72215830, #72215846, #72232796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271743);