Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #94871 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×#97 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
796 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)(3,4)(6,8)(9,10,13,19,14,16,22,23,15,11,17,20,12,18,21,24);(2,4)(10,16,11,18)(13,17)(19,24,20,23)(21,22);(5,6,7,8)(9,11,13,20,14,18,22,24,15,10,17,19,12,16,21,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,c^b=ch,f^b=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239251, #72239254, #72239256, … |
| maximal normal subgroup of | #72239251, #72239254, #72239256, … |
| maximal quotient of | #72239317, C2×#147372, #72322514 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54568);