Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501164 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1052 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)(3,7)(4,6)(5,8)(10,14)(11,17)(15,22)(16,21)(18,24)(20,23);(1,3,4,8)(2,5,6,7)(12,19)(18,24);(9,10,12,14)(11,15,18,20)(13,16,19,21)(17,23,24,22);(2,6)(5,7)(9,11,13,17)(10,15,16,23)(12,18,19,24)(14,20,21,22)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72285788, #72285863, #72285962, … |
| maximal normal subgroup of | #72285788, #72285863, #72285962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409914);