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 | |
5303 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(18,23)(19,25)(26,30)(27,32);(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,26)(22,23)(24,25)(27,28)(29,30)(31,32);(2,4)(6,7)(10,14)(12,15)(21,29)(26,30)(27,32)(28,31);(3,8)(6,7)(10,14)(11,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27);(1,4,5,2)(3,7,8,6)(9,12,13,15)(10,11,14,16)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8305754);