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 | #36182 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4807 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)(3,4,6,7)(10,13)(14,16)(17,18)(19,24)(20,30)(21,31)(22,25)(23,26)(27,28)(29,32);(1,3)(2,4)(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,23,29)(18,24,26,32)(20,22,31,27)(21,28,30,25);(1,4)(2,6)(3,8)(5,7)(17,20,23,31)(18,21,26,30)(19,27,29,22)(24,25,32,28);(1,3)(2,7)(4,8)(5,6)(9,11,12,15)(10,14,13,16)(17,20)(18,21)(19,22)(23,31)(24,28)(25,32)(26,30)(27,29);(1,3)(2,4)(5,6)(7,8)(17,21)(18,20)(19,28)(22,24)(23,30)(25,29)(26,31)(27,32);(1,4)(2,6)(3,8)(5,7)(17,22)(18,25)(19,20)(21,32)(23,27)(24,30)(26,28)(29,31)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7875217);