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 | |
4862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,6)(4,8)(9,10,13,12)(11,15,16,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,30)(31,32);(1,2)(3,8)(4,6)(5,7)(9,11,13,16)(10,14,12,15)(18,23)(19,27)(25,28)(26,32);(1,3)(2,4)(5,6)(7,8)(9,12,13,10)(11,15,16,14)(17,19,22,27)(18,24,23,29)(20,26,30,32)(21,28,31,25);(1,2)(3,8)(4,6)(5,7);(1,4,5,8)(2,6,7,3)(17,20)(18,25)(19,26)(21,24)(22,30)(23,28)(27,32)(29,31);(1,2)(3,4)(5,7)(6,8)(17,21)(18,26)(19,28)(20,29)(22,31)(23,32)(24,30)(25,27)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=g,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7764920);