Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,7)(9,10,13,16,14,17,12,21)(11,18,20,15,19,24,22,23);(1,2)(3,4)(5,6)(7,8);(2,4)(5,7,8,6)(9,11)(10,15)(12,22)(13,20)(14,19)(16,24)(17,23)(18,21);(11,19)(15,23)(18,24)(20,22)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^c=di,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72237203, #72237217, #72237222, … |
| maximal normal subgroup of | #72237203, #72237217, #72237222, … |
| maximal quotient of | #72238773, #72238776, #72239076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265102);