Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
6562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(6,9)(7,8)(13,17)(15,16)(21,22,26,24)(23,29,31,27)(25,32,33,28)(30,34,36,35);(1,2)(3,4)(5,6,11,9)(7,14,8,12)(10,17,18,13)(15,19,16,20);(5,7,11,8)(6,12,9,14)(10,15,18,16)(13,19,17,20)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,8,11,7)(6,12,9,14)(10,16,18,15)(13,19,17,20)(21,24,26,22)(23,27,31,29)(25,32,33,28)(30,34,36,35);(5,9,11,6)(7,12,8,14)(10,17,18,13)(15,19,16,20)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35);(5,10,11,18)(6,13,9,17)(7,15,8,16)(12,19,14,20)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7651413);