Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470122 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,29,32)(20,26,22,25)(24,30,31,28);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,23)(24,31)(25,26)(27,32);(11,15)(13,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(11,15)(13,16)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,27,31,32);(10,14)(13,16)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377646);