Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#97 |
| prime radical | C24 | #97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(10,14,15,20)(11,17)(16,23,22,21)(18,24);(9,10,11,16,12,14,18,21,13,15,17,22,19,20,24,23);(2,5)(6,8);(10,15)(14,20)(16,22)(21,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,b^a=bh,c^a=cg,f^a=fi,d^b=di,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72227016, #72286868, #72287094, … |
| maximal normal subgroup of | #72227016, #72286868, #72287094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387158);