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 | |
6882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6,11,15)(7,14,17,9)(8,16,19,12)(10,13,18,20)(24,32)(27,33)(31,35)(34,36);(1,2)(3,4)(5,7)(6,9)(8,13)(10,16)(11,17)(12,18)(14,15)(19,20)(21,22)(23,30)(24,33)(25,28)(26,29)(27,32)(31,34)(35,36);(5,8)(6,12)(7,13)(9,18)(10,14)(11,19)(15,16)(17,20)(21,23,26,28)(22,25,29,30)(24,31,32,35)(27,34,33,36);(5,9,11,14)(6,7,15,17)(8,18,19,10)(12,13,16,20)(21,24)(22,27)(23,31)(25,34)(26,32)(28,35)(29,33)(30,36);(5,10,11,18)(6,13,15,20)(7,16,17,12)(8,14,19,9)(21,22)(23,25)(24,27)(26,29)(28,30)(31,34)(32,33)(35,36);(5,7)(6,14)(8,13)(9,15)(10,12)(11,17)(16,18)(19,20)(21,25)(22,28)(23,29)(24,34)(26,30)(27,35)(31,33)(32,36)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7566115);