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 | |
4734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(4,8)(17,18,22,26)(19,21,25,23)(20,24,30,28)(27,31,32,29);(2,6)(3,7)(9,10)(11,14)(12,15)(13,16)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,29,28,31);(1,2,5,6)(3,4,7,8)(9,11)(10,13)(12,16)(14,15);(2,6)(3,7)(9,10)(11,13)(12,15)(14,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,31)(29,32);(1,3)(2,4)(5,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28);(1,4)(2,7)(3,6)(5,8)(9,10)(11,13)(12,15)(14,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9993864);