Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518936 |
| 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 | |
1364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,16)(13,15)(19,26)(20,29)(22,31)(23,27)(24,30)(28,32);(1,3,5,4)(2,6,8,7)(9,11,14,10)(12,16,13,15)(17,18)(19,27)(20,30)(21,25)(22,32)(23,26)(24,29)(28,31);(10,11)(15,16)(17,19,20,26,21,28,29,32)(18,22,24,31,25,23,30,27);(9,12,14,13)(10,15,11,16)(17,19,20,26,21,28,29,32)(18,23,24,27,25,22,30,31);(1,4,5,3)(2,7,8,6)(9,13,14,12)(10,16,11,15)Polycyclic presentation:
2^9|a^2=fghi,b^2=f,c^2=h,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=cgi,d^b=dh,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511768);