Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #59409 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(4,7)(10,13)(14,16)(17,18,23,25)(19,24,27,31)(20,22,30,32)(21,28,29,26);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,19,23,27)(18,24,25,31)(20,28,30,26)(21,22,29,32);(2,6)(3,8)(18,25)(19,27)(20,30)(21,29);(17,20)(18,22)(19,28)(21,31)(23,30)(24,29)(25,32)(26,27);(1,3)(2,7)(4,6)(5,8)(17,21,23,29)(18,26,25,28)(19,22,27,32)(20,31,30,24);(1,4,5,7)(2,3,6,8)(9,11,12,15)(10,14,13,16)(17,22)(18,20)(19,29)(21,27)(23,32)(24,28)(25,30)(26,31)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,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, 7616597);