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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1460 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)(10,14)(11,12)(17,18)(19,24)(20,26)(21,31)(22,32)(23,28)(25,30)(27,29);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19,23,20)(18,24,28,26)(21,22,30,29)(25,27,31,32);(9,11,13,12)(10,15,14,16)(17,20,22,30,23,19,29,21)(18,25,27,26,28,31,32,24);(9,11,13,12)(10,16,14,15)(17,21,22,20,23,30,29,19)(18,26,27,31,28,24,32,25);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,14,16)Polycyclic presentation:
2^9|a^2=fghi,b^2=fi,c^2=g,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,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7500260);