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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,15)(12,13)(14,16)(17,18,23,27)(19,22,21,24)(20,26,31,25)(28,32,29,30);(1,3)(2,6)(4,7)(5,8)(10,13)(14,15)(17,19,23,21)(18,24,27,22)(20,29,31,28)(25,30,26,32);(9,10)(11,14)(12,13)(15,16)(17,20)(18,25)(19,28)(21,29)(22,30)(23,31)(24,32)(26,27);(1,4,5,6)(2,3,7,8)(9,11,12,16)(10,14,13,15)(17,21,23,19)(18,22,27,24)(20,29,31,28)(25,30,26,32);(1,4)(2,3)(5,6)(7,8)(17,20)(18,26)(19,29)(21,28)(22,30)(23,31)(24,32)(25,27);(1,3)(2,4)(5,8)(6,7)(17,22)(18,19)(20,30)(21,27)(23,24)(25,28)(26,29)(31,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10319242);