Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500450 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1683 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)(3,6,7,8)(18,24)(19,28)(21,22)(25,30)(26,27)(31,32);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,27,23,26,21,24)(19,29,28,31,30,20,25,32);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,28,23,30,21,25)(18,20,27,32,26,29,24,31);(11,16)(13,15)(17,20,23,29)(18,25,26,28)(19,24,30,27)(21,31,22,32);(10,14)(13,15)(18,26)(20,29)(24,27)(31,32)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413798);