Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
972 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)(9,10,13,17,14,18,12,15)(11,19,21,24,22,23,20,16);(2,5)(6,8)(10,15,18,17)(11,20,22,21)(16,24)(19,23);(1,3,4,7)(2,6,5,8)(9,11,12,20,14,22,13,21)(10,16,17,19,18,24,15,23);(2,5)(6,8)(10,15,18,17)(16,23,24,19)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=e,d^2=fh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72338916, #72338922, #72338944, … |
| maximal normal subgroup of | #72338916, #72338922, #72338944, … |
| maximal quotient of | #72238813, #72238814, #72238815 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286467);