Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C22×D4 | #41 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10,13,17,14,18,22,16)(11,19,12,15,20,23,21,24);(9,11,13,12,14,20,22,21)(10,15,16,19,18,24,17,23);(2,5)(3,8)(6,7);(9,12,13,20,14,21,22,11)(10,15,17,23,18,24,16,19)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=gi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264814, #72265057, #72307197, … |
| maximal normal subgroup of | #72264814, #72265057, #72307197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 118913);