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 | |
6771 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)(5,6,10,12)(7,11,16,13)(8,9,15,14)(17,19,20,18)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(1,2,3,4)(5,7)(6,11)(8,17)(9,19)(10,16)(12,13)(14,18)(15,20);(6,12)(7,16)(8,15)(18,19)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(5,7)(6,13)(8,17)(9,18)(10,16)(11,12)(14,19)(15,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,8)(6,14)(7,17)(9,12)(10,15)(11,18)(13,19)(16,20)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,9,10,14)(6,15,12,8)(7,18,16,19)(11,17,13,20)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7882168);