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 | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10,12,14)(11,13,16,15)(17,18,22,23)(19,20,24,28)(21,29,30,25)(26,32,31,27);(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32);(1,3,5,8)(2,4,6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,24)(21,30)(27,32);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,32)(30,31);(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,29)(30,32);(17,21)(18,25)(19,26)(20,27)(22,30)(23,29)(24,31)(28,32)Polycyclic presentation:
2^9|a^2=h,c^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7589283);