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 | #36318 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,7)(10,13)(14,16)(19,25)(20,27)(23,29)(24,28);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(1,2)(3,7)(4,5)(6,8)(18,22)(23,29)(24,28)(30,32);(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,31)(28,32);(1,3,4,8)(2,6,5,7)(17,18)(19,23)(20,28)(21,22)(24,27)(25,29)(26,32)(30,31);(1,2)(3,7)(4,5)(6,8)(17,20)(18,24)(19,26)(21,27)(22,28)(23,30)(25,31)(29,32)Polycyclic presentation:
2^9|d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7571861);