Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #253363 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
940 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)(3,5)(4,7)(6,8)(12,19)(14,20)(15,22)(18,24);(3,8)(5,6)(10,14,16,20)(11,17)(15,23,22,21)(18,24);(2,5)(3,8)(6,7)(9,10)(11,15)(12,20)(13,16)(14,19)(17,22)(18,23)(21,24);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)Polycyclic presentation:
2^9|b^2=e,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,e^c=eh,f^c=fi,g^c=gh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72283599 |
| maximal normal subgroup of | #72283599 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 160559);