Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343406 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | #62805 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7,6,3,4,8)(10,15)(11,13)(16,22)(17,21)(18,20)(23,24);(1,3)(2,6)(4,8)(5,7)(9,10,14,17)(11,15,13,21)(12,16,19,23)(18,22,20,24);(1,4,6,5)(2,7,3,8)(9,11,14,13)(10,15,17,21)(12,18,19,20)(16,22,23,24);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24);(12,19)(16,23)(18,20)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,f^2=i,g^2=hi,b^a=bf,c^a=ci,f^a=fh,f^b=fi,g^b=ghi,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251300);