Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2078 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)(3,4)(5,6,8,7)(9,10)(11,14)(12,19)(13,18)(15,17)(16,21)(20,23)(22,24);(2,4)(9,11,13,10,15,18,14,17)(12,19,21,16,22,24,20,23);(5,7,8,6)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24);(10,17)(11,18)(16,23)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72238844, #72239936, #72239938 |
| maximal normal subgroup of | #72238844, #72239936, #72239938 |
| maximal quotient of | #72238846, #72239947 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261150);