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 | #34715 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,3)(4,7,8,6)(10,13)(14,16)(19,26)(24,30)(27,32)(28,31);(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,31)(28,32);(2,3)(6,7)(9,11,12,15)(10,14,13,16)(18,23)(20,29)(24,30)(27,32);(1,3,5,2)(4,7,8,6)(17,19)(18,24)(20,27)(21,28)(22,26)(23,30)(25,31)(29,32);(1,4)(2,6)(3,7)(5,8)(17,20,22,29)(18,25,23,21)(19,27,26,32)(24,31,30,28);(17,21,22,25)(18,20,23,29)(19,28,26,31)(24,27,30,32)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,e^2=i,f^2=i,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, 7895185);