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 | |
4898 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16);(1,3,5,4)(2,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,21)(22,23)(25,30)(26,32)(27,29)(28,31);(1,2,5,8)(3,6,4,7)(18,23)(21,25)(24,31)(27,32);(1,2)(3,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,31)(25,32)(26,30);(1,4,5,3)(2,7,8,6)(17,20)(18,25)(19,26)(21,23)(22,30)(24,27)(28,29)(31,32);(17,21,22,25)(18,20,23,30)(19,27,28,32)(24,29,31,26)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7706722);