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 | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,17,14,18,12,16)(11,15,19,22,20,24,21,23);(2,5)(6,8)(9,11,13,19,14,20,12,21)(10,15,17,22,18,24,16,23);(1,3,4,7)(2,6,5,8)(10,16,18,17)(15,22,24,23);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72306970, #72307343, #72307377 |
| maximal normal subgroup of | #72306970, #72307343, #72307377 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275401);