Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #512070 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,8,4,6)(10,14,15,22)(11,17)(12,20)(16,23,21,19)(18,24);(1,3,5,4)(2,6,7,8)(11,18)(14,22)(17,24)(19,23);(11,18)(14,22)(17,24)(19,23);(1,4,5,3)(2,6,7,8)(9,10,12,21,13,15,20,16)(11,19,17,14,18,23,24,22)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=eh,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419545);