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 | #44882 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4696 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)(3,4)(5,7)(6,8)(10,13)(14,16)(17,18,23,20)(19,27,28,24)(21,25,31,30)(22,29,32,26);(1,3,6,5)(2,4,8,7)(9,10,12,13)(11,14,15,16)(17,19,23,28)(18,24,20,27)(21,26,31,29)(22,25,32,30);(1,3,6,5)(2,7,8,4)(19,28)(21,31)(24,27)(25,30);(1,4,6,7)(2,3,8,5)(9,11,12,15)(10,14,13,16)(17,20,23,18)(19,27,28,24)(21,30,31,25)(22,29,32,26);(1,3,6,5)(2,4,8,7)(17,21)(18,25)(19,29)(20,30)(22,24)(23,31)(26,28)(27,32);(1,5,6,3)(2,7,8,4)(17,22,23,32)(18,26,20,29)(19,25,28,30)(21,24,31,27)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=hi,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, 7587839);