Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682500 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(17,18,21,23)(19,27,28,32)(20,26,22,25)(24,29,31,30);(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,15,12,14)(18,23)(19,28)(25,26)(29,30);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,19,20,29,21,28,22,30)(18,24,25,32,23,31,26,27);(1,3,4,7)(2,6,5,8)(17,20,21,22)(18,25,23,26)(19,29,28,30)(24,32,31,27);(9,12,13,11)(10,15,16,14)(19,28)(24,31)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fghi,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6761720);