Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437428 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1702 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)(3,6)(4,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,26)(21,27)(22,31)(23,25)(24,29)(30,32);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,20)(18,24,25,32)(21,29,22,30)(26,31,28,27);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(18,25)(24,32)(26,28)(27,31);(1,4,5,3)(2,7,8,6)(11,15)(13,16)(17,20,23,19)(18,26,25,28)(21,29,22,30)(24,31,32,27);(17,21,23,22)(18,27,25,31)(19,29,20,30)(24,26,32,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366120);