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 | |
5170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,29)(21,26)(22,23)(24,28)(25,30)(31,32);(1,3,7,6)(2,4,8,5)(10,13)(14,15)(18,23)(20,30)(24,27)(26,31);(1,4,7,5)(2,6,8,3)(9,10)(11,14)(12,13)(15,16);(1,4,7,5)(2,3,8,6)(9,11)(10,14)(12,16)(13,15)(17,19)(18,24)(20,31)(21,29)(22,28)(23,27)(25,32)(26,30);(1,5,7,4)(2,3,8,6)(17,20)(18,25)(19,26)(21,24)(22,30)(23,29)(27,32)(28,31);(1,6,7,3)(2,4,8,5)(17,21)(18,26)(19,25)(20,24)(22,32)(23,31)(27,30)(28,29)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=h,e^2=h,f^2=h,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, 10318931);