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 | |
4419 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)(5,8)(6,7)(10,12)(14,15)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28);(1,3)(2,8)(4,6)(5,7)(9,10,13,12)(11,15,16,14)(19,27)(21,31)(23,32)(25,28);(1,2,7,6)(3,8,5,4)(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30);(1,4)(2,5)(3,6)(7,8)(9,12,13,10)(11,15,16,14);(1,5,7,3)(2,8,6,4)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32);(1,6,7,2)(3,8,5,4)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9148137);