Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,7)(9,10,12,11)(13,14,19,21)(15,24,16,27)(17,23,18,22)(20,26,28,25);(2,4)(5,7)(6,8)(9,11)(10,12)(13,15,19,16)(14,20,21,28)(17,26,18,25)(22,27,23,24);(6,10)(7,11)(9,12)(13,16,19,15)(14,20,21,28)(17,25,18,26)(22,27,23,24);(14,21)(20,28)(22,23)(24,27)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140572);