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 | |
5775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,12)(13,14)(15,16)(17,18)(19,27)(20,25)(21,29)(22,23)(24,28)(26,31)(30,32);(1,2,5,6)(3,7,8,4)(10,14)(11,16)(18,23)(19,28)(25,32)(26,29);(1,2)(3,7)(4,8)(5,6)(11,16)(12,15)(18,23)(19,28)(20,30)(21,31);(2,6)(3,8)(9,11,13,16)(10,15,14,12)(17,19)(18,24)(20,26)(21,25)(22,28)(23,27)(29,30)(31,32);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,25)(19,29)(21,27)(22,30)(23,32)(24,31)(26,28);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,21,22,31)(18,26,23,29)(19,25,28,32)(20,27,30,24)Polycyclic presentation:
2^9|b^2=i,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8968792);