Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #343075 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2679 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #18674 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,7,4,8,5,6)(10,16,18,24)(11,19,22,21);(1,3,4,5)(2,6,8,7)(9,10)(11,20)(12,19)(13,24)(14,16)(15,18)(17,21)(22,23);(2,7)(3,5)(6,8)(9,11,15,22)(10,17,18,12)(13,19,14,21)(16,23,24,20);(2,7)(3,5)(6,8)(9,12)(10,11)(13,23)(14,20)(15,17)(16,21)(18,22)(19,24)Polycyclic presentation:
2^9|a^2=ef,b^2=hi,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 250281);