Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1267 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,13,12)(11,14,16,15)(18,23)(19,28)(20,22)(24,32)(25,29)(30,31);(2,5)(6,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,29)(20,26)(21,23)(22,27)(24,28)(25,30)(31,32);(9,12,13,10)(11,15,16,14)(17,19,20,28,21,31,22,30)(18,24,27,29,23,25,26,32);(11,16)(14,15)(17,19,20,28,21,31,22,30)(18,25,27,32,23,24,26,29);(1,3,4,8)(2,6,5,7)(17,20,21,22)(18,26,23,27)(19,30,31,28)(24,29,25,32)Polycyclic presentation:
2^9|a^2=fghi,c^2=hi,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511325);