Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #334695 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #19671 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,7,5,4,3,8)(9,10)(11,16)(12,21)(13,23)(14,19)(15,20)(17,22)(18,24);(1,3)(5,6)(7,8)(9,11,15,13)(10,16,20,23)(12,18,17,19)(14,21,24,22);(1,3,5,6)(2,7,4,8)(9,12,15,17)(10,11,20,13)(14,23,24,16)(18,22,19,21);(1,4,5,2)(3,8,6,7)(9,13,15,11)(10,17,20,12)(14,21,24,22)(16,19,23,18)Polycyclic presentation:
2^9|a^2=eh,b^2=i,c^2=hi,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gi,f^c=fi,g^c=gi,e^d=eh,f^d=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 242002);