Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115542 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10)(11,17,19,14)(12,20)(13,15)(16,21)(18,23,22,24);(1,3)(2,6)(4,8)(5,7)(11,18)(12,21)(14,23)(16,20)(17,24)(19,22);(1,4,6,5)(2,7,3,8)(9,11)(10,14)(12,22)(13,19)(15,17)(16,24)(18,21)(20,23)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,f^b=fi,d^c=di,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72316663, #72316671 |
| maximal normal subgroup of | #72316663, #72316671 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22743);