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 | |
4626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,12)(13,14)(15,16)(17,18,23,25)(19,24,28,32)(20,21,26,31)(22,30,29,27);(1,2,5,8)(3,4,7,6)(10,14)(11,15)(17,19,23,28)(18,24,25,32)(20,29,26,22)(21,27,31,30);(1,3)(2,6)(4,8)(5,7)(9,11,13,15)(10,12,14,16)(18,25)(19,28)(20,26)(27,30);(1,4)(2,7)(3,8)(5,6)(9,12)(10,15)(11,14)(13,16)(17,20)(18,21)(19,29)(22,28)(23,26)(24,27)(25,31)(30,32);(1,3)(2,4)(5,7)(6,8)(17,21)(18,26)(19,30)(20,25)(22,32)(23,31)(24,29)(27,28);(1,4)(2,7)(3,8)(5,6)(17,22,23,29)(18,27,25,30)(19,20,28,26)(21,24,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,f^2=h,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, 10003688);