Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,23)(18,24);(1,2)(3,6)(4,8)(5,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,24,23)Polycyclic presentation:
2^9|d^2=i,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,f^c=fhi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 144581);