Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10)(11,21,22,24)(12,23,16,17)(13,18)(14,19)(15,20);(1,3,4,8,5,2,6,7)(9,11,14,23,15,22,13,17)(10,16,19,24,20,12,18,21);(9,12,15,16)(10,17,20,23)(11,18,22,19)(13,24,14,21);(2,3)(7,8)(9,13,15,14)(10,18,20,19)(11,17,22,23)(12,24,16,21)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=di,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239519, #72338786, #72338838 |
| maximal normal subgroup of | #72239519, #72338786, #72338838 |
| maximal quotient of | #72239509 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 101033);