Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #47123 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,7,4,6)(17,18,22,26)(19,25,21,23)(20,28,29,24)(27,31,30,32);(1,3,5,4)(2,6,8,7)(10,13)(11,15)(17,19,22,21)(18,23,26,25)(20,27,29,30)(24,31,28,32);(1,3,5,4)(2,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,29)(18,24,26,28)(19,27,21,30)(23,31,25,32);(9,11,12,15)(10,14,13,16)(20,29)(24,28)(27,30)(31,32);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,21,22,19)(18,25,26,23)(20,30,29,27)(24,32,28,31);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608932);