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 | #59137 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,4)(3,7,5,8)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32);(1,3,6,5)(2,7,4,8)(9,11,13,16)(10,14,12,15)(18,22)(23,25)(24,28)(30,31);(2,4)(7,8)(17,18,21,22)(19,25,27,23)(20,24,29,28)(26,31,32,30);(1,4,6,2)(3,8,5,7)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31);(9,10,13,12)(11,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,5,6,3)(2,8,4,7)(9,12,13,10)(11,15,16,14)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=i,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7585460);