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 | |
6803 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(21,22,27,24)(23,29,25,28)(26,30,34,32)(31,36,33,35);(1,2)(3,4)(5,6)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(21,23,27,25)(22,28,24,29)(26,31,34,33)(30,35,32,36);(6,10)(7,13)(12,15)(14,19)(22,24)(28,29)(30,32)(35,36);(5,7,9,13)(6,11,10,17)(8,14,16,19)(12,18,15,20)(21,24,27,22)(23,29,25,28)(26,32,34,30)(31,36,33,35);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,25,27,23)(22,29,24,28)(26,33,34,31)(30,36,32,35);(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,26,27,34)(22,30,24,32)(23,31,25,33)(28,35,29,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7643304);